Ultrasonic reflection and material evaluation

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Ultrasonic reflection and material evaluation

W.G. Mayer

To cite this version:

W.G. Mayer. Ultrasonic reflection and material evaluation. Revue de Physique Appliquée, Société française de physique / EDP, 1985, 20 (6), pp.377-381. �10.1051/rphysap:01985002006037700�. �jpa- 00245347�

Ultrasonic reflection and material evaluation

W. G. Mayer

Physics Department, Georgetown University, Washington ^D.C. 20057, ^U.S.A.

(Reçu ^{le 7} novembre, accepté le 9 novembre 1984)

Résumé. 2014 On donne dans l’article ^un panorama des principes physiques ^{de base} concernant la réflectivité ultra-

sonore en relation avec la caractérisation de matériaux solides. Les exemples ^{donnés se} rapportent à la réflexion

critique, ^{aux ondes de} Rayleigh ^{et de Lamb et à} l’orientation cristalline effectuée au moyen de ^mesures de reflexion.

Abstract ²⁰¹⁴ An overview is given describing ^the physical basis of ultrasonic reflectivity ^as it relates to solid material characterization. The examples given illustrate critical angle reflection, Rayleigh ^{and Lamb waves, and} crystal

orientation via reflection measurements.

Classification

Physics ^Abstracts

43.20

1. Introduction.

The reflection coefficient of ultrasonic waves for a flat

boundary

^formed

by

^{a solid and a fluid}

depends

very

strongly

^{on the}

angle

^{of incidence} and the mechanical

properties

of the materials involved. ^In

particular, reflectivity

^shows very

pronounced

maxima and minima as well as other

changes

^{at critical}

angles

^of

incidence. Critical

angles

^{are defined as those}

angles

of incidence

(in

^the

fluid)

^{where a refracted wave}

(in

the solid) ^travels

parallel

^{to the}

boundary

^between

the media as indicated in

figure

^{1. The} angle ^{of refrac-}

tion for the

longitudinal

^{wave, a, or} for the shear wave,

j8,

^{in the} solid is in

general

^{related -10 the} angle

of incidence, 0, ^{in the}

liquid by

^{Snells’ law}

sin

03B8/vf

^{= sin}

03B1/v1

^{= sin}

03B2/vs, (1)

Fig. ^{1. -} (a) ^{Snells’ law} refraction for ultrasonic waves at infinite half _space boundary. (b) Conversion to plate ^modé

vibration.

where vf

is the sonic

velocity

in the fluid and _v,

and v.

are the velocities of the

longitudinal

^{and shear} waves,

respectively,

in the solid. Therefore, the critical angle

of incidence for the

longitudinal

wave,

6L,

^{will be}

defined as the

particular

⁰ for which a ⁼ 900 in

equation (1); correspondingly Os

^{is defined as the 0}

for

which P

^{= 900 in}

equation (1).

Snell’s law

only yields

^the

angles

of reflection and refraction but not the amount of

reflectivity

^{or trans-}

missivity

into the solid These latter

quantities

^were

first calculated

by

^Knott [1], ^a

seismologist,

^{in 1899}

for

plane

^waves.

Rewriting

his results

yields

^{a reflec-}

tion coefficient for the energy which is of the form

where

The

density

of the solid is

given by

p and that of the fluid

by

03C1f.

An examination of

equation (2)

shows that if 0

= 03B8L

^{or 0 =}

Os,

the ratio of reflected to incident energy is

unity

while for all other

angles

^{of incidence}

(R/I)2

is less than

unity

except ^{when 0} >

Os

^{in which}

case total reflection occurs. Based on this

simple analysis

^{one can make use} of reflection measurements to determine a number of

properties

^{of the}

reflecting

solid.

Although

there exist a great ^number of variants,

only

the fundamental features of three distinct combi-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01985002006037700

378

nations of parameters will be discussed here. These three _groups are : fluid-solid boundaries (both ^media

are considered to be infinite half

spaces) ;

^solid plates

in fluids;

anisotropic

solid-fluid

boundary.

2. Infinite half _space boundaries.

When the

reflecting

solid is thick enough ^{so that the}

product frequency (of

the incident ultrasound, ⁱⁿ MHz) ^times sample ^thickness

(in mm)

^is greater ^than

approximately

15, ^one can, to ^a very

good approxi-

mation, ^use

equation (2)

^to

experimentally

^{locate a}

reflection

peak

^at

03B8L.

^{If the}

sample

is immersed in water

(vf

^{= 1.49}

km/s)

then the reflection coefficient

yields

information about the

longitudinal velocity

^of

the solid. Plots of the reflection coefficient ^are shown in

figure

^{2 for a ceramic material}

(alpha-silicon- carbide, v1

^{= 11.76}

km/s, vs

^{= 7.5}

km/s),

^a

typical

metal

(brass, vl

^{= 4.43}

km/s, Vs

^{= 2.12}

km/s),

^{and a}

plastic (plexiglas, v1

^{= 2.67}

km/s, vs

^{= 1.12}

km/s).

^It

is seen that in all cases the reflection

peak

^{at the}

longitudinal

^critical angle ^{is very}

pronounced

^and

well defined,

making

^it

possible

^to

determine v, simply by observing experimentally

^{the angular loca-}

tion of the reflection

peak

^-

eliminating

^{the need}

to propagate ultrasound through ^the sample.

Fig.2. ^{- Classical} reflection coefficient for boundary

between water and plexiglas (solid line), ^brass (dash-dot),

and alpha ^{silicon carbide} (dotted line).

It should be noted that the curves in

figure

^{2 show}

a

sharp

^{rise to 100}

% reflectivity

^{at the shear wave}

critical

angle,

except ^for

plexiglas

^{for which no real}

Os

^exists

because vf

> Vs and therefore sin 0 >

sine

in

equation (1).

Some of the earliest

experimental investigations

^of

critical

angle reflectivity

[2] ^{confirmed the}

general validity

^{of the} theory. ^The

03B8L peak

^is

relatively

easy to detect

although

^{it does not}

usually

reach 100

%

due to the fact that attenuation is not considered [3]

in

equation (2).

However, ^the

predicted

total reflec- tion for

Os (when

^{a real}

Os exists)

^is

ordinarily

^unob-

servable. Instead, ^near

Os

^the

reflectivity

^decreases

[2]

to values of 50

%

^{or less at a 0}

slightly

greater ^than

Os.

Just as

0L

^is reached when oc becomes 900, ^{this new}

critical angle is reached when another

possible

^wave

travels along ^the surface. A reflection

dip

^occurs

whenever this _wave, a

Rayleigh-type

wave, is _gene- rated. Rayleigh waves are surface waves whose

velocity

^{is less than the shear wave}

velocity,

^{and in}

a first approximation ^this

velocity

^is given [4]

by

where J is the Poisson’s ratio of the solid.

One could add one more term to Snells’ law, equa- tion

(1), namely ...

^{= sin}

y/vr,

^but since this

Rayleigh-

type ^wave only ^travels

along

^the surface, ^sin y ⁼ 1

always, giving

^{rise to} another critical angle ^{of inci-} dence. This critical

angle, usually

^{called the}

Rayleigh angle, OR,

denotes the incident

angle

under which the

Rayleigh-type

^{wave is}

generated.

^{This wave}

travels

along

the interface and radiates its energy back into the

liquid

^{with the} angle of radiation

being equal

^{to the}

angle

of reflection. The maximum of radiation is

occasionally

^«

displaced »

along ^the

surface as shown in

figure

^{3a. This} phenomenon ^of

nonspecular

reflection was first observed

by

^Schoch [5]

for

Rayleigh angle

^incidence.

Nonspecular

reflections were later

investigated by

Bertoni and Tamir [6] for ultrasonic beams of finite width incident at ^the

Rayleigh

angle ^{and later}

by

Ngoc et ^al. [3] ^{for other} angles,

showing

^{the existence}

of nonspecular reflection effects other than a

simple

«

displacement

^».

Among

^{them the most noticible}

effect is a

split-up

^{of an} incident ultrasonic beam into a

double-peaked

reflected beam as shown as an

example ⁱⁿ

figure

^3b.

The occurrence of nonspecular reflection at critical

angles ^is useful in material characterization,

parti- cularly

for surface

homogeneity

determinations. Once the

Rayleigh angle

^{of a} material has been determined

[through

^{use of}

equation (3)

^{or other}

formulae]

^and

the transducer has been

positioned

^{over the interface}

so that

Rayleigh angle nonspecular

reflections are

observable, the reflected beam should not

change

^its

characteristic beam

profile

^{when the}

sample

^{is moved}

parallel

^{to itself} If, however, any

portion

^{of the}

sample

surface has an

inhomogeneity

^which

changes

the elastic

properties,

^and

thus v1

and v., the

Rayleigh

wave

velocity

^also

changes

in this isolated location.

This in turn makes the local value of v, different from the rest of the sample and the fixed incident beam no

longer

^{strikes at the}

Rayleigh angle

when the local

inhomogeneity

is irradiated. As a consequence, the

nonspecular

reflection

profile

changes. ^This

change

in the beam

profile

^can

easily

^{be seen} in Schlieren

images

^{similar to}

figure

^{3. Local}

changes

^{in the}

value

of vr

^{of about 1}

%

^{can be detected and an}

inhomogeneity

^can thus be localized.

Thus,

reflectivity

^{studies can} reveal small local

inhomogeneities.

^Another

application

^of

reflectivity

studies is concerned with the determination of the thickness of a

plate

^one side of which _may not be accessible. This aspect is discussed in the next section.

Fig. ^{3. -} Examples ^of nonspecular reflection. (a) ^lateral

shift with location of reflected beam indicated by ^dashed

lines if reflection would be specular, (b) double-peaked nonspecular reflection.

3. Solid plates ^{in a floid}

When a flat, homogeneous ^solid plate ^is

subjected

^to

an

external locally applied periodic

^{force the}

plate

may be in resonance with the force and the disturbance _may propagate along ^the plate. ^{It was shown}

by

^Lamb [7]

that more than one resonance is

possible

^{and thus one}

may observe different normal modes of vibration,

depending

^{on the} thickness of the

plate,

its elastic _pro-

perties,

^{and the}

frequency

of excitation. The

defining equations

^{for Lamb} modes, ^{in their}

simplest

forms,

are given

by

4 LS coth (03C0Sfd/w) -

(1 ⁺

S 2)2 coth (nLfd/w) = -

^iX

4 LS tanh (03C0Sfd/w) -

(1 ⁺

S2)2 tanh(Lfd/w) = - iX

(4)

where

The unknown, the Lamb mode

velocity,

^{w, has more}

than one

possible

value ; ^the

possible

magnitudes ^of

w

depend

very much on the

product fd,

^{the bulk wave}

velocities and to a lesser extent on the ratio of the

liquid

and solid densities.

The

applied

^force may well be an ultrasonic beam

striking

^the plate ^{which is} immersed in a fluid as

indicated in

figure

^{1 b. If}

03BBp

^{is the}

wavelength

^{of one}

of the possible plate modes there will be a

correspond- ing

0 that will match the wavefronts at the interface and a plate mode will be

generated

When this condi- tion is met the reflection coefficient for the incident

beam, ^{in a first} approximation, goes ^{to zero.} However, if a bounded ultrasonic beam is used

nonspecular

reflection effects become

important,

^{similar to the case}

of

Rayleigh

angle incidence for infinite half _{spaces. In} the case of solid plates ^{there will be a} number of critical

angles ^of incidence,

corresponding

^{to the}

generation

of the various plate ^modes (or ^Lamb modes). ^The computation of the velocities of plate ^{modes is some-}

what involved [8], and the calculations are

quite

^cum-

bersome [9] ^when incident bounded ultrasonic beams

are considered. The use of the results, ^{on the other} hand, ^is

quite simple.

^{As an} illustration of a

technique

to determine the thickness of a plate ^when only ^one

side of the plate ^is accessible, ^consider

figure

^{4. It}

represents ^a portion of the Lamb mode

velocity

^dis- persion ^{curves for a brass} plate immersed in water.

The abscissa is labelled

fd (in

MHz. mm) ^{and the}

ordinate in terms of Lamb wave velocities and in

angle

of incidence in water, ^{the two}

being

^related through Snells’ law.

Fig. ^{4. -} Section of velocity dispersion ^{curves for Lamb}

modes.

Assume now that a 1 mm thick brass plate ^{is used}

and that the ultrasonic beam has a

frequency

^{of 4 MHz.}

This locates the

product fd at

4, ^{the same as for a} 4 mm

brass plate ^{and a}

frequency

^{of 1 MHz. One notes that}

Lamb modes can exist when the angle ^is 46-60, 35.8°, 24.4°, ^{18.3° or 15.9°} because the possible ^{Lamb mode}

velocities are 2.05, 2.55, 3.6, ^4.75 and 5.45

km/s.

^Under

these conditions reflections should be nonspecular.

Some modes are easier to excite than others and the form of the nonspecular reflection effects _may well differ even for the same beam

being

used, ^{with atte-}

nuation of the modes and beam width

being

^the

determining

^factors [10].

Suppose ^{now that a small} section of the plate ^{is 5}

%

thinner than the rest of the plate. ^In this thin section

380

the value of

fd is

^{reduced to 3.8 and the} mode velocities increase except ^{the lowest} velocity ^mode (which ^is equal ^{to the}

Rayleigh

mode). ^{Thus the}

previously

determined Lamb angles ^{are now} different. If such a

plate ^is scanned, ^{with the} angle of incidence fixed so

that a distinct nonspecular reflection

profile

^{is obser-} vable, ^the nonspecular ^character of the reflection will

collapse ^{into a} pure specular reflection

profile

^when

the thin portion ^{of the} plate is irradiated.

Fortunately,

^{the same}

principle

holds when the

plate

^is

asymmetrically

^loaded (i.e., ^{water on one side}

and another fluid or a gas ^on the other).

Although equation (4)

would have to be

expanded

^{to include}

the second fluid the

velocity dispersion

^curves usually

vary little from those for the

symmetrically

^{loaded case.}

Thus one can find the thickness of the

plate

^{even if}

only

^one side is accessible. It is evident that one can

detect corroded or otherwise thinned sections of containers of hazardous

liquids,

^{or one can determine}

the thickness of a plate ^{which cannot be}

easily

^reached

from both sides, ^{or the} presence of flaws which inhibit the normal

propagation

^{of Lamb} modes, ^etc.

4. Crystal orientation.

In the

previous

sections the assumption ^{was made}

that the solid is isotropic. ^The situation becomes much

more

complicated

^when

anisotropic (single crystal)

substances are considered. In this case the bulk wave

velocities are orientation

dependent

^{as was shown}

by

Green [11] ^as

early

^{as 1839. Just as the}

Rayleigh

^velo-

city

ⁱⁿ

isotropic

substances

depends

^on the values of vl

and vs [Eq. (3)],

^{so does the}

Rayleigh

^wave

velocity

on the surface of a

single crystal,

except ^here v, is also orientation

dependent.

Comprehensive ^search

methods were

employed by

^Farnell [12] ^to prepare surface wave

velocity

^{curves for}

single

crystals.

Figure

⁵ shows the incident angle ^{in water needed}

to excite a surface wave (outer portion ^of

Fig.

5) ^{or a}

pseudo-surface

^wave [12] ^{on the}

(100)-face of copper.

One notes that the incident angle ^{has an extremum}

when one of the surface waves is to be

propagated

ⁱⁿ

the

[110]-direction

^{on the}

(100)-face.

This fact allows

one to find this direction on the

crystal

face and thus all the other

crystallographic

directions

simply by making

^use

again

^of nonspecular reflection pheno-

mena. If the incidence is

adjusted

^to

correspond

^to

the

Rayleigh angle

^{or the}

pseudo-surface

^{wave critical}

angle

^{for the}

[110]-direction,

^a

nonspecular

effect will

occur

only

^{when the}

projection

of the incident beam onto the sample surface is in the

[110]-direction.

Fig. ^{5. - Variation of critical} angle incidence for a water- copper (100)-plane interface, ^{based on data} of reference [12].

Rotation of the sample under the incident beam will thus show

nonspecular

reflection only when the line- up between the incident beam and the

[110]-direction

is satisfied ^- thus the crystal has been oriented It should be

pointed

^{out that no}

theory

exists which will predict

precisely

^what type ^of nonspecular ^reflec-

tion will occur when different beam widths and fre-

quencies

^are used in this

application.

Nevertheless, ^the

method has been used

successfully

[13] for the orien- tation

of crystals

and for the angle

of energy

^flow [14]

in surface waves on

single crystals.

5. Conclusioa

Nonspecular reflectivity

of ultrasonic beams is

always

connected with wave excitation at a critical

angle.

Observation of these

phenomena yields

information about these critical

angles

^{and thus enables one to}

determine

important properties

^{of the} solid, ^{as for}

instance wave velocities,

inhomogeneities,

changes ⁱⁿ

thickness and certain

crystal

orientation aspects. ^The

topics

discussed above are illustrations of some of the

possibilities

^to characterize solids

by using

^ultrasonic

reflection measurements.

Acknowledgments,

The work on which

ihis

_paper is based has been

supported by

^{the Office of Naval Research, U.S.} Navy.

References

[1] KNOTT, ^C. G., ^Philos. Mag. ⁴⁸ (1899) ^64.

[2] ROLLINS, ^F. R., Material Eval. 24 (1966) ^683.

[3] NGOC, T. D. K. and MAYER, W. G., ^J. Appl. Phys. ⁵⁰ (1979) ^7948.

[4] BERGMANN, L., ^Der Ultraschall, 6th ed. (Hirzel Verlag, Stuttgart) ^1954.

[5] SCHOCH, A., Acustica 2 (1952) ^19.

[6] ^BERTONI, H. L. and TAMIR, T., Appl. Phys. ^{Lett. 2} (1973) ^157.

[7] ^LAMB, H., ^{Proc. R. Soc.} London, A ⁹³ (1917) ^114.

[8] VIKTOROV, ^I. A., Rayleigh and Lamb Waves (Plenum Press, ^New York) ^1967.

[9] NGOC, T. D. K. and MAYER, ^W. G., IEEE Trans.

SU-27 (1980) ^229.

[10] NGOC, T. D. K. and MAYER, W. G., IEEE Trans.

SU-29 (1982) ^112.

[11] GREEN, G., ^Trans. Cambr. Philos. Soc. 7 (1839) ^121.

[12] ^{FARNELL, G. N., in} Physical Acoustics, ^{W. P.} Mason, ed., ^{vol. 6} (Academic Press, ^New York) ^1970.

[13] DIACHOK, ^{O. I. and} MAYER, W. G., Acustica 26 (1972)

267.

[14] BEHRAVESH, ^{M. and} MAYER, W. G., IEEE Ultras.

Symp. ^{Proc. 1974, p. 104.}