elastic wave propagation and ndt

8/8/2019 Elastic Wave Propagation and Ndt

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Proc. Natl. Sci. Counc. ROC(A)

Vol. 23, No. 6, 1999. pp. 703-715

(Invited Review Paper)

Elastic Wave Propagation and NondestructiveEvaluation of Materials

TSUNG-TSONG WU

Institute of Applied Mechanics

National Taiwan University

Taipei, Taiwan, R.O.C.

(Received April 13, 1999; Accepted June 7, 1999)

ABSTRACT

The use of elastic waves to measure elastic properties as well as flaws in solid specimens has received

much attention, and many important applications have been developed recently. This paper summarizes

some of the recent results in applying elastic waves to nondestructive evaluation (NDE) of isotropic as

well as anisotropic materials. This paper is divided into four parts, the first part describes the theoretical

background of surface waves in an anisotropic layered medium, where sextic formalism of surface waves

is adopted. The second part introduces numerical simulations of 2-D and 3-D transient elastic waves

propagating in plate structures that contain cracks and/or flaws. In the third part, some of the applications

of laser generated surface waves used to determine anisotropic elastic constants and bonding conditions

in layered media are described. In the last part, a newly developed method for determining concrete elastic

constants and recently developed methods for detection of surface breaking cracks in concrete using

transient elastic waves are described.

Key Words: elastic wave, nondestructive evaluation, anisotropy, crack, concrete

703

I. Introduction

The use of elastic waves to measure elastic prop-

erties as well as flaws of solid specimens has received

interest; for example, in the use of elastic waves in

nondestructive evaluation of concrete structures, in the

use of laser generated ultrasonic waves in the deter-

mination of anisotropic elastic constants of composite

materials and in the recovery of the bonding properties

and/or thickness of bonded structures. In the above

applications, the analyses can generally be divided intothree parts: forward simulation of elastic wave

propagation, elastic wave measurement and inverse

analysis (or signal processing). Forward simulation of

elastic waves includes finding various analytical so-

lutions (exact or approximated) of some simple geom-

etry structures and numerical solutions using finite

difference, finite element or boundary element methods.

Elastic wave measurement involves using various wave

sources (steel ball impact, pencil break, laser heating

etc.) and receivers (piezo-ceramics sensors, laser in-

terferometry etc.). Inverse analysis or signal analysis

of elastic waves involves using efficient inverse algo-

rithms, imaging processing, neural networks etc.

In the development of an efficient elastic wave

based nondestructive evaluation (NDE) method, the

first step is understanding the propagation character-

istics of elastic waves. In the literature, there exist

many theoretical analyses of transient elastic wave

propagation in simple structures which are necessary

to understand elastic wave physics. However, to fur-

ther study waves diffracted from an arbitrary obstacle,

numerical techniques are of great importance. In this

paper, some of the recent results obtained in applying

surface wave dispersion and transient elastic waves tothe measurement of the elastic properties of anisotropic

layered media as well as in-situ concrete crack and

elastic constants are summarized. The sextic formal-

ism of surface waves in anisotropic multi-layer media

and the numerical solution of transient elastic wave

propagation are also introduced.

II.Surface Waves in AnisotropicSolids

1. Anisotropic Half Space

The early studies on the propagation of elastic


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T.T. Wu

704

surface waves in crystals can be found in a review

article by Farnell (1970). Basically, most of the studies

on anisotropic surface wave problems adopted the for-

mulation utilized by Synge (1957); therefore, iterations

between the secular equations of the equations of motion

and the complex surface boundary determinants were

needed to determine the surface wave velocity. Aformal mathematical proof for the existence of surface

waves in all directions in an anisotropic material was

given by Barnett and Lothe (1974). Their proof was

based upon the works of Stroh (1962) in the develop-

ment of the theory of dislocations. In the Stroh

formalism, the material properties and elastic symme-

tries are contained in the so-called fundamental elas-

ticity tensor (Chadwick and Smith, 1977). The solution

of a particular problem can be expressed in terms of

the eigenvalues and eigenvectors of a six-dimensional

tensor of rank two (Lothe and Barnett, 1976). Barnett,

Lothe and their coworkers (Barnett et al., 1973a,1973b)extended the Stroh formalism to the integral formalism.

They showed that the surface wave velocity in an

anisotropic material can be determined by vanishing

the determinant of a real symmetric 22 matrix without

solving the eigenvalue problem encountered in the

Stroh formalism.

Below is an example of anisotropic surface wave

velocities calculated using the Stroh-Barnetts integral

formalism. The three nonvanishing elastic constants

of the single Silicon crystal are C11=165.7 GPa, C12=

63.9 GPa and C44=79.56 GPa, and the density is 2332

Kg/m3. The propagation of surface waves on the (001)

and (110) surfaces was considered.

Figure 1 shows the variation in surface wave

velocity along different propagation directions on the

(001)-plane. We note that for propagation along the

[110] direction, the surface wave velocity is equal tothe limiting velocity, and this is the so-called excep-

tional surface wave velocity.

2. Layered Anisotropic Half Space

Theoretical analyses of the propagation of surface

acoustic waves in layered media have been reported

in the literature. Ewing et al. (1957) reviewed the early

analyses of the dispersion of surface waves in an iso-

tropic layered medium. In the last decade, the appli-

cations of acoustic microscopy and fiber reinforced

composites have initiated interest in studying wave

propagation in layered isotropic or anisotropic media

(Kundu and Mal, 1986; Nayfeh and Taylor, 1988;

Bouden and Datta, 1990). Experimental and inverse

analyses of surface waves in an anisotropic medium

or a layered medium have also been reported (Chai and

Wu, 1994; Wu and Liu, 1999).

In addition to the conventional formulation of

surface wave propagation in layered media (Ewing et

al ., 1957), the sextic formalism of Stroh (1962) has

recently been employed. In the sextic formalism, the

equations of motion and the constitutive equations are

combined and arranged to form a first-order matrix

differential equation. The displacement and the trac-tion acting across the planes normal to the layering

surfaces are grouped into a six-dimensional vector. In

each layer, the solution of the ordinary differentiated

equations forms a transfer matrix that can be utilized

to map the variables from one surface to the next

layering surface. With this formulation, the rank of

the matrix encountered in the computation is indepen-

dent of the number of layers. The following example

is calculated using the sextic formalism. Shown in Fig.

2 is the dispersion of the Rayleigh wave and Love wave

in a single layered anisotropic half space. The aniso-

tropic half space is a single crystal Silicon with the(001) surface normal, and the top layer is SiO2 . The

propagation direction is along the [100] direction.

3. Layered Anisotropic Half Space Loaded withViscous Liquid

Recently, the development of micro-acoustic wave

sensors in biosensing (Andle and Vetelino, 1994) has

created the need for further investigation into surface

wave propagation in a fluid loaded layered medium.

Kim (1992) investigated the effect of an adjacent viscous

fluid on the propagation of Love waves in a layered

half space medium. His results include an exact so-

Fig. 1. The variation of the surface wave velocity along different

propagation directions on the (001)-plane.


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Elastic Wave and NDE of Materials

705

lution and an asymptotic solution for the velocity and

attenuation, which are expressed in terms of the vis-

cosity and density of the viscous fluid. Zhu and Wu

(1995) presented a theory for the propagation of Lamb

waves in a plate bordered by a viscous liquid. Their

results also include numerical solutions of the Lamb

wave dispersion related to sensing applications. A

detailed experimental study on a Love wave sensor for

biochemical sensing in liquids was reported by Kovacs

et al. (1994). They showed that, for small viscosity,

the interaction of an acoustic Love wave with a viscous

liquid can be described by a Newtonian liquid model.

The substrates (layered half space) of the above men-

tioned investigations are assumed to be isotropic.

However, anisotropic layered substrates may appear in

many practical applications. Recently, Wu and Wu1

studied the propagation of Rayleigh and Love waves

in a viscous loaded anisotropic layered half space

medium. The sextic formalism of Stroh (Stroh, 1962;

Braga, 1990) was extended to obtain the viscous liquideffect on the dispersion relation of surface waves. The

dispersion relation was expressed in terms of the acoustic

impedance of the anisotropic substrate and the viscous

liquid.

In the following example, the propagation of

surface waves in an anisotropic single-layered half

space loaded with a viscous liquid is considered. (The

anisotropic half space is silicon, and the surface layer

is SiO2, 5 m in thickness.) Figure 3 shows the cal-culated results for the fundamental Rayleigh wave

propagating on the [001] surface along the direction

15 away from the [100] axis.

III. Numerical Simulation of Transient

Elastic Waves

1. Finite Difference Formulations

For a body with dynamic disturbance propagating

in a three dimensional space, the stress equations of

motion can be written in the Voigts form as

v it

= Tij, j + fi , (1)

where is the mass density, fi is the body force perunit mass, Tij represents the stress components, v i

represents the particle velocities and i,j=1, 2, 3. Fora linear elastic isotropic medium, the Hookes law

reads as

Tijt

= vk, kij +(v i, j + vj, i) , (2)

where and are the Lam constants.To study elastic wave propagation in a general two

or three-dimensional heterogeneous medium, the finite

difference scheme with staggered grids can be adopted

(Virieux, 1986). In the heterogeneous formulation ,

instead of treating the internal interfaces using explicit

interfacial boundary conditions, changing the elasticconstants and mass density can be adopted. The finite

difference formulae for the free surface grids are derived

Fig. 2. The dispersion of the Rayleigh wave and Love wave in a

single layered anisotropic half space.

Fig. 3. The phase velocity of surface waves in a anisotropic single-

layered half space loaded with a viscous liquid.

1Wu, T. T. and T. Y. Wu, Surface waves in coated anisotropic medium loaded with viscous liquid. Submitted.


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T.T. Wu

706

based on the method of a fictitious line (Alterman andKaral, 1968). For the details of the finite difference

formulation of 2-D and 3-D elastic wave propagation

problems, the readers is referred to the works of Virieux

(1986), Wu and Gong (1993), Wu and Fang (1997b)

and Fang and Wu (1997).

2. Numerical Simulation of 2-D DiffractedWaves

Shown in Fig. 4(a)-(c) are snapshots (t=6, 12, 18

s) of diffracted wave fields induced by a steel ball

drop in an aluminum plate which contains a normalsurface-breaking crack. The depths of the crack and

the source to crack opening distances are the same and

are equal to 2 cm, and the height of the plate is 8 cm.

In Fig. 4(a) (t=6 s), the Rayleigh surface waves propa-gating in both directions can be identified clearly while

the P wave-front has crept around the crack tip. In Fig.

4(b) (t=12 s), the Rayleigh surface wave which propa-gates toward the crack has crept down the crack. The

Rayleigh surface wave reflected from the corner of the

top of the crack can also be observed. In Fig. 4(c) (t=18

s), the Rayleigh wave has crept around the crack tip,and the creeping P and S waves have carried wave

energy to the shadow zone located across the surface

opening.

Shown in Fig. 5(a)-(c) are snapshots (t=6, 12, 18

s) of diffracted wave fields for the case of an alumi-

num plate with a circular hole located at mid-depth.The diameter of the hole is 2 cm. In Fig. 5(a) (t=6

s), one can find that the P wave front is just approach-ing the top edge of the hole. In Fig. 5(b) (t=12 s),the P wave front has passed the hole, and the S-wave-

front is at the top of the hole. In addition, scattering

of the P wave by the hole is observed clearly. In Fig.

5(c) (t=18 s), a creeping wave has formed and hascrept down along the boundary of the hole with a certain

amount of energy.

We note that the complicated elastic wave fields

scattered by an obstacle in a structure can be observed

more clearly by means of numerical simulated snapshots.

Snapshots of scattered wave fields can be processed

to obtain a dynamic image of the scattered wave field

to observe the time evolution of the various elastic

wave modes.

3. Numerical Simulation of 3-D DiffractedWaves

In this subsection, we will discuss the numericalFig. 4. Snapshots of the diffracted wave fields generated by a normal

surface breaking crack.

Fig. 5. Snapshots of the diffracted wave fields generated by a cir-cular hole.

(a)

(b)

(c)

(a)

(b)

(c)


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707

simulation of 3-D diffracted waves induced by surface-

breaking cracks in concrete. Figure 6 shows that the

depth of the crack is d, and that the distances from the

source and the receiver to the surface opening of the

crack are a and b, respectively. Figure 6 also

shows the diffracted waves received at different re-

ceiver distances ofb=2, 4, 6, 8 cm. The longitudinalwave and the transverse wave velocities of the concrete

are assumed to be 4200 m/s and 2450 m/s, respectively.

The source distance a=3 cm, the depth of the crackd=6

cm and the source time function is a half sin3/2 twith

a contact duration equal to 20 s. From Fig. 6, we notethat the amplitude of the diffracted wave decays with

the receiver distance; in addition, the initial slope of

the diffracted wave-front is also dependent on the

receiver distance. It has been shown that the phase

changes of the initial signal are dependent on the relative

angle formed between the lines of the source to crack

tip and the receiver to crack tip (Wu et al., 1995a).The 3-D finite difference formulation can also be

utilized to simulate elastic wave diffraction due to

cracks with different cross sections. Shown in Fig. 7

is the computed B-scan image of a normal surface

breaking crack with a rectangular cross section. The

cross section of the rectangular crack is 20 cm in length

and 5 cm in depth. Shown in Fig. 8 is the computed

B-scan image of a normal surface breaking crack with

a semi-elliptical cross section. The surface opening

length of the semi-elliptical crack is 20 cm, and the

maximum depth is 5 cm. In the figures, the horizontal

axis is the elapsed time, and the vertical axis is the test

position. In the images, the amplitude of the received

signal is characterized by the black and white contrast,

i.e., the bigger the amplitude, the darker the image; the

smaller the amplitude, the brighter the image. From

the B-scan images, one can observe the differences ofthe crack cross sections easily.

IV. Nondestructive Evaluation ofMaterials Using Laser GeneratedSurface Waves

Due to its noncontact feature and ability to gen-

erate broadband signals, laser ultrasonics has demon-

strated great potential in NDE applications (Scruby and

Drain, 1990). Laser generated ultrasonic waves have

been applied to investigate the Lamb wave propagation

phenomena in thin plates (Hutchins et al., 1989;

Dewhurst et al., 1987; Nakano and Nagai, 1991) and

to obtain scan images of thin graphite/epoxy laminates

and silicon wafer (Veidt and Sachse, 1994). LaserFig. 6. The diffracted waves received at different receiver distances

b=2, 4, 6, 8 cm. [Adopted from Wu and Fang (1997b)]

Fig. 7. The computed B-scan image of a normal surface breaking

crack with a rectangular cross section. [Adopted from Wu

and Fang (1997b)]

Fig. 8. The computed B-scan image of a normal surface breaking

crack with a semi-elliptical cross section. [Adopted from Wuand Fang (1997b)]


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T.T. Wu

708

generated ultrasonic bulk waves has also been used to

determine the elastic constants of anisotropic materials

(Castagnede et al., 1991) and subsurface defects (Cho

et al., 1996).In recent years, the ultrasonic point-source/point-

receiver (PS/PR) technique has also been shown to be

a convenient method for measuring group velocities

(Scruby, 1989; Kim et al., 1993). The acoustic waves

generated by a pulsed laser beam or a breaking capillary

propagate in all directions and can be detected by small

aperture sensors located on the surface of a specimen

or, if available, by interferometric sensors. The po-

sitions of the source and receiver determine the direc-

tion of the measured group velocities (Every and Sachse,

1990; Doyle and Scala, 1991; Kim and Achenbach,

1992).

1. Determination of Anisotropic Elastic Con-stants

In the case of isotropic media, only one longitu-

dinal velocity and one shear wave velocity are needed

to determine the Lames constants and . The pro-cedure for determining the elastic constants of aniso-

tropic solids from measured phase velocities of ultra-

sonic bulk waves has been studied in detail by Every

(1980). However, the wave energy is propagated with

the group velocity, not the phase velocity, and the phase

velocity generally is different from the group velocityboth in direction and magnitude. Every and Sachse

(1990) have used the PS/PR technique to measure the

group velocities of both longitudinal and shear waves,

and furthermore, to determine the elastic constants of

anisotropic solids from measured group velocities. As

pointed out by these authors, the recovery of elastic

constants from phase velocity measurements is straight-

forward and well established, but the same is not true

for group velocities because no general closed-form

expression is available to relate the elastic constants

to the group velocities, and because it is difficult to

distinguish between the slow transverse and fast trans-verse group velocities in a given direction especially

in folded regions. Doyle and Scala (1991) determined

the elastic constants for composite overlays by employ-

ing a line focused pulsed laser. In their scheme, both

the phase velocities and the skew angles between the

phase and group velocities of longitudinal waves skim-

ming the surface are measured to fit the elastic constants.

The aforementioned studies on determining the elastic

constants of anisotropic solids depended on bulk wave

velocity measurements.

Kim and Achenbach (1992) determined the elastic

constants of coating films by using a line focus scan-

ning acoustic microscope (SAM) to measure the dis-

persion curves of surface waves propagating along

some symmetry axes. Mendiket al. (1992) used the

Brillouin scattering technique and SAM to measure

surface wave velocities and determined the elasticconstants of single crystal nickel. The aforementioned

studies on recovering elastic constants were based on

surface wave phase velocity measurements, and the

conventional surface wave theory developed by Synge

(1957) was utilized in the forward calculations. We

note that when the conventional surface wave theory

is employed to determine the surface wave phase ve-

locities of anisotropic solids, iterations between the

secular equations of the equations of motion and the

complex surface boundary determinants are needed.

Only when the phase velocity of a surface wave is

determined can the group velocity be calculated using

a straightforward substitution. The iteration process

in the aforementioned formulation is complicated, and

there is no priori way to know whether surface wave

solutions exist or not. Therefore, to determine the

elastic constants of an anisotropic solid inversely from

surface wave velocity measurements, an efficient for-

mulation for the forward calculation of the surface

wave velocity is needed.

Instead of using the conventional method, Wu and

Chai (1994) applied the Stroh-Barnett integral formal-

ism to calculate numerically the surface wave phase

and energy velocities in a unidirectional fiber-rein-

forced composite. In their study, experiments were alsoconducted to measure the energy velocities of surface

waves using laser-generated ultrasound and the PS/PR

technique. The results showed that the calculated and

experimental slowness curves of the surface wave energy

velocity were in good agreement. In a subsequent

paper, Chai and Wu (1994) further proposed a method

for determining anisotropic elastic constants based on

surface wave energy velocity measurements. The

method has three parts: a laser ultrasound experiment,

forward calculation of the surface wave phase and

energy velocities, and an inversion algorithm for re-

covering the anisotropic elastic constants or the crystalorientation. Figure 9 shows a polar plot of the measured

laser ultrasonic signals of a composite specimen, and

Fig. 10 shows a comparison between the calculated and

the measured slowness curves of the energy velocities

of the surface waves. Figure 11 shows a polar plot of

the measured signals of a silicon crystal with [111]

orientation. Figure 12 shows a comparison between

the calculated and the measured surface wave velocities.

2. Inverse Determination of the Bonding LayerProperties

In a recent work by Wu and Chen (1996), the


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dispersion of laser generated surface waves in an epoxy

bonded copper-aluminum layered specimen was studied.

In their study, a laser ultrasonic experiment based on

the PS/PR technique was conducted to measure surface

wave signals in a layered specimen. An Nd:YAG laser

was utilized as a point source, and elastic wave signals

were received using a piezoelectric transducer with a

small acting area. The received wave signals were then

processed in the frequency domain to obtain the dis-

persion relation of the fundamental surface wave mode.

In a subsequent study, Wu and Liu (1999) employed

an inversion algorithm to determine the thickness and

the elastic properties of a bonding layer from the mea-

sured dispersion relation of laser generated surface

waves. A computer program for calculating the phase

velocity dispersion of general isotropic and/or aniso-

tropic layered media was utilized to explore the influ-ence of the epoxy-bonded layer. The dependence of

the error function on the inversion parameters, such as

the thickness, elastic wave velocities and the density

of the bonding layer, was studied first. Inversions of

the bonding layer thickness and the elastic wave ve-

locities of the epoxy layer were then performed. The

results showed that under thickness or elastic property

inversions, only one global minimum existed in each

of the inversion problems. The inversion results dem-

Fig. 9. A polar plot of measured laser ultrasonic signals of a com-

posite specimen. [Adopted from Chai and Wu (1994)]

Fig. 10. A comparison between the calculated and the measured

slowness curves of the energy velocities of surface waves.

[Adopted from Chai and Wu (1994)]

Fig. 11. A polar plot of the measured laser ultrasonic signals of a

single crystal silicon. [Adopted from Chai and Wu (1996)]

Fig. 12. A comparison between the calculated and the measured

slowness curves of the energy velocities of surface waves.[Adopted from Chai and Wu (1994)]


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T.T. Wu

710

onstrated that the thickness or the elastic properties of

the bonding layer could be successfully determined.

Figure 13 shows the measured phase velocity disper-

sion (stars) and the calculated phase velocity dispersion

(lines) of the fundamental mode of surface waves in

a layered specimen with different bonding thicknesses.

3. Detection of Unbond Region in LayeredStructures

In a recent paper, Wu and Chen (1999) utilized

the wavelet transform to study the dispersion of laser

generated surface waves in an epoxy bonded copper-

aluminum layered specimen with and without unbond

areas. Laser ultrasonic experiments based on the PS/

PR technique were undertaken to measure the surface

wave signals in a layered specimen. The wavelet

transform with a Morlet wavelet function was adoptedto analyze the group velocity dispersion of the surface

wave signals. A novel hybrid formula for group velocity

dispersion was proposed for wave propagation across

unbond regions. The results and data obtained were

in good agreement with the calculated and the experi-

mental dispersion curves. The general behavior of the

group velocity dispersion for different measurement

configurations could potentially be utilized to differ-

entiate unbond regions in a layered structure.

Figure 14 shows the theoretical calculations of the

group velocity dispersions for the case where l1=35

mm, l2=70 mm and the bonding thickness varies from

0 to 0.2 H. l1, l2 are the propagation distances in the

unbond region and the well bond region, respectively.

The theoretical predictions show that the initial parts

of the dispersion curves are dominated by the funda-

mental mode of the anti-symmetric Lamb wave. Theresults shown in Fig. 14 demonstrate that the proposed

group velocity dispersion (with a bonding thickness of

0.1 H) for measurements across the unbond region

agrees well with the measured data (solid circles).

V. Nondestructive Evaluation ofConcrete Structures

The increasing need for monitoring of structure

integrity has motivated intensive research into the de-

velopment of nondestructive testing and evaluation

methods for civil infrastructures, especially for con-

crete structures. Although concrete has been used for

many decades, it still is one of the most widely used

building materials. It can be found in bridges, buildings,

foundations, dams, highways etc. Unlike metallic

materials that are usually assumed to be homogeneous,

concrete consists of cement, sand and aggregates of

different sizes. In addition, many microcracks or voids

in concrete due to the manufacturing process or exter-

nal loading often appear. The above mentioned factors

strongly affect the physical behavior of concrete. For

example, the existence of voids or microcracks in

concrete decreases its compressive strength.

In the case of nondestructive evaluation of con-crete structures, diffraction and attenuation of elastic

waves in concrete are pronounced. A relatively low

frequency (large wave-length), high energy source is

Fig. 13. The measured phase velocity dispersion (stars) and the

calculated phase velocity dispersion (lines) of surface waves.

[Adopted from Wu and Liu (1999)]

Fig. 14. The group velocity dispersions for the case where l1=35mm and l2=70 mm. [Adopted from Wu and Chen (1999)]


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711

usually adopted to generate elastic waves in concrete.

A successful technique using transient elastic waves

in nondestructive evaluation of concrete structures is

the Impact-Echomethod. This method was developedand applied by Carino, Sansalone and their coworkers

(Carino et al., 1986; Sansalone and Carino, 1988; Lin

et al., 1990; Carino and Sansalone, 1992) in the de-

tection of flaws or voids in concrete slab and wall struc-

tures. In this method, a steel ball impacting the surface

of a concrete structure is employed to generate elastic

waves. A displacement point receiver is placed next

to the impact source to receive the wave signals. The

Fast Fourier Transform (FFT) is adopted in the Impact-

Echo method for analysis of the received wave-form.

The periodic reflection of the longitudinal wave be-

tween the top surface and the flaw induces a dominant

peak in the frequency spectrum, and if the correspond-

ing frequency is termed f, then the depth of the flaw,

D, can be determined from the formula D=VL/2f.

The elastodynamic scattering problems have

played an important role in the quantitative nondestruc-

tive evaluation of materials (Achenbach et al., 1980).

Their solutions can be applied to develop inverse scat-

tering techniques for flaw sizing and to model the

reliability of flaw detection. There are analytical,

numerical and experimental studies on transient elastic

wave scattering induced by cracks in metallic struc-

tures (Kundu and Mal, 1981; Scandrett and Achenbach,

1987; Paffenholz et al., 1990; Liu and Datta, 1993; Wuet al., 1995a). The aforementioned studies showed that

the waves diffracted from surface-breaking cracks offer

sufficient information to measure the crack geometry.

In addition, to extract more information about a crack,

waveform analysis of the measured diffracted wave

was suggested. Wu and Fang (1997b) and Fang and

Wu (1997) successfully applied the inversion tech-

nique to determine the depth of a surface-breaking

crack in concrete. Both normal and oblique surface-

breaking cracks can be detected using their method.

Due to the large aperture of ultrasonic transducers,

ultrasonic measurement of cracks in complicated con-crete structures is not easy (if indeed possible). The

point source/point receiver method can eliminate this

limitation and can be further utilized to obtain an image

of the crack profile (Lin and Su, 1996; Liu et al., 1996).

In the following, recent results of the measure-

ment of elastic constants and/or the depth of surface-

breaking cracks in concrete structures using transient

elastic waves are summarized.

1. Measurement of the Concrete Elastic Con-stant

The elastic constants of concrete are usually

determined by using a uniaxial test to determine the

Youngs modulus of a standard cylindrical specimen.

In the nondestructive evaluation of concrete, the ultra-

sonic wave velocity is usually measured and utilizedto predict the strength of the concrete (Ben-Zeitun,

1986). Wu et al. (1995b) proposed a method for de-

termining the dynamic elastic constants of a concrete

specimen using transient elastic waves. In their study,

the Rayleigh wave velocity was determined using the

cross correlation method and the longitudinal wave

velocity, which was determined by measuring longi-

tudinal wave-front arrival. The major limitation in

measuring the longitudinal wave velocity of a concrete

specimen using the aforementioned method is the re-

quirement that a perpendicular corner be present. As

a longitudinal wave travels across a corner, it produces

a larger displacement jump at the point of longitudinal

wave-front arrival when a conical transducer (Proctor,

1982) (vertically polarized and normal to the specimen

surface) is used. Furthermore, in Rayleigh wave ve-

locity measurement, the relatively long distance from

source to receiver hinder the broader application of the

method proposed by Wu et al. (1995b).

In a subsequent paper, Wu and Fang (1997a) re-

moved the limitations of the aforementioned method

(Wu et al., 1995b) and proposed a new method based

on the measurement of horizontally polarized surface

responses. In their new method, the longitudinal wave-

front can be identified directly from the surface response.In addition, the source to receiver distance employed

to determine the Rayleigh wave velocity is reduced.

In their study, for the purpose of detecting the radial

component of a transient elastic wave signal, two

horizontally polarized conical transducers were

fabricated. The tip of the conical element was about

1.5 mm in diameter. In their experiment, they con-

firmed the correctness of the aforementioned theoreti-

cal analyses. With the Rayleigh wave and the longi-

tudinal wave velocities known, the transverse wave

velocity of the concrete specimen could be obtained

by solving the well-known Rayleigh wave equation.Figure 15 shows the vertical (solid line) and horizontal

(broken line) components of the surface wave signal

generated by a steel ball impacting the origin of the

half space. The points of arrival of the longitudinal

wave (P), transverse wave (S) and Rayleigh wave (R)

are indicated in Fig. 15. We note that the arrival of

the Rayleigh wave does not coincide exactly with the

first dip of the vertical component (solid line) of the

wave response. The arrival of the longitudinal wave-

front is relatively small when compared with the rest

of the signal. In addition, the transverse wave arrival

is continuous at the vertical displacement signal. In

contrast to the vertical component, the horizontal


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T.T. Wu

712

component (broken line) of the longitudinal wave shows

a much clearer displacement jump at the point of wave-

front arrival. Furthermore, the arrival of the Rayleigh

wave induces a sharp corner and can be identified

easily.

The influence of the reinforced bars on the de-

termination of longitudinal wave and Rayleigh wave

velocities in reinforced concrete structures has also

studied by Wu et al. (2000). They utilized the finite

difference method to study elastic wave scattering in

reinforced concrete specimens and, further, to examinethe influence on the elastic wave velocity measure-

ments of transient elastic waves. The parameters that

may affect Rayleigh wave velocity measurements, such

as the cover thickness H and the spacing of the

reinforcement, were considered. Numerical results

showed that the previously proposed method for mea-

suring the elastic wave velocity of plain concrete can

be applied to the case of reinforced concrete with

reasonable accuracy. We note that if the reinforce-

ments in a concrete structure are allocated (for example,

using a rebar locator) first, then the accuracy of the

transient elastic wave method can be increased.

2. Concrete Crack Depth Measurements

Measurement of the size and geometry of a sur-

face-breaking crack is important in evaluating an ex-

isting concrete structure because the existence of sur-

face-breaking cracks in a concrete structure decreases

its durability significantly. The position of a surface-

breaking crack can be easily observed with the naked

eye; however, nondestructive determination of the depth

of the crack is not trivial. The inhomogeneous nature

of concrete structures prevents the utilization of high

frequency ultrasonics. The use of transient elastic

waves, based on the PS/PR technique, has received

much interest in the field of nondestructive evaluation

of concrete structures. For the detection of cracks in

concrete, Wu and Fang (1996) utilized the phase char-acteristics of the diffracted wave front to determine the

depth of normal surface-breaking cracks. Measure-

ment of the arrival of the diffracted wave-front was

utilized by Lin and Su (1996) to determine the depth

of a normal surface-breaking crack and by Liu et al.

(1996) to construct images of surface-breaking cracks.

Using the finite difference forward solution and a

nonlinear optimization algorithm, the geometry of a

surface-breaking crack in concrete could be determined

inversely using the measured diffracted wave signal

(Wu and Fang, 1997b; Fang and Wu, 1997). The

inverse results showed that both the depth and the

inclined angle of an oblique surface-breaking crack

could be accurately determined.

The principle of migration in reflection seismol-

ogy can be combined with transient elastic wave mea-

surements to construct the image of a surface-breaking

crack in a concrete structure (Liu et al., 1996). Con-

sider a cracked half-space, and let a signal be emitted

from a source on the surface. The signal is diffracted

by the crack tip and arrives at a receiver which is on

the opposite side of the crack opening. Suppose the

travel time of the diffracted signal is t, and that the

wave velocity is v; then, the travel distance of the signal

is vt. If the diffraction path is unknown, any point inthe medium with the same travel distance is a possible

diffraction point. Therefore, the crack tip should fall

on an ellipse with the source and receiver as its foci.

The exact diffraction point can generally be located by

finding the intersection of two different ellipses (ob-

tained using different source and receiver arrangements).

The inversion of diffracted waves method utilizes

time domain diffracted wave signals to inversely de-

termine the crack depth of a surface breaking crack in

concrete (Wu and Fang, 1997b; Fang and Wu, 1997).

In this method, a steel ball is used to generate the elastic

wave, and a receiver located across the crack receivesthe diffracted elastic wave. A three dimensional finite

difference program is used to calculate the forward

solution of the diffracted wave. With the forward

numerical solution found, the initial part of the mea-

sured diffracted wave signal can then be utilized to

recover the crack depth using a stable optimization

algorithm.

VI. Concluding Remarks

It is worth noting that successful development of

an elastic wave based nondestructive evaluation method

for testing materials depends strongly on a solid un-

Fig. 15. The surface vertical displacement due to a steel ball

impacting the origin of a half space. [Adopted from Wu

and Fang (1996)]


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713

derstanding of wave propagation characteristics. In

addition to theoretical analysis, many elastic wave

related problems require utilization of a general-pur-

pose computer program for calculating elastic wavepropagation in structures with non-simple boundaries

(including layered structures).

As for the application of laser ultrasonics to the

nondestructive evaluation of materials, the recent re-

sults presented in this paper are very encouraging. This

approach can be utilized to determine the elastic con-

stants of anisotropic materials, the bonding properties

of bonded structures etc. In addition, the application

of surface acoustic wave (SAW) devices in biosensing

and micro-electro-mechanical system (MEMS) re-

searches has again triggered the study of surface waves.

For measuring the elastic constants or the unifor-

mity of reinforced concrete structures, the recently

proposed method that is based on the measurement of

skimming longitudinal wave and Rayleigh wave ve-

locities has been shown to be robust for in situ meas-

urement. For detecting the depth of a surface-breaking

crack in concrete, the transient elastic wave methods

described in this paper have also been shown through

both the numerical and experimental analyses to be

feasible.

Acknowledgment

The author thanks the National Science Council of theR.O.C. for financial support under grants NSC 86-2621-P-002-045

and NSC 86-2212-E-002-078.

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