Journal of Nondestructive Evaluation, VoL 12, No. 2, 1993

A New Technique for Time-Domain Ultrasonic NDE of Thin Plates Changyi Zhu I and V i k r a m K. Kinra I

Received March 14, 1992; revised September 25, 1992

A new time-domain ultrasonic NDE technique is reported in this paper for the measurement of the thickness (given the wavespeed) or wavespeed (given the thickness) of thin plates; by thin we mean the thickness is less than the wavelength. By introducing a retrieve function the incident field can be reconstructed from the transmitted (reflected)field by a two-term (three-term)summation. A systematic sensitMty analysis of this technique has been carried out. The new technique has been used to measure the thickness or the wavespeed of aluminum plates with thickness ranging from 0.089 to 6.426 mm using (low frequency) 1-MHz transducers: 0.014<(thickness/wavelength)_<1.0. The measurement error was found to be about 1% for thickness and 3% for the wavespeed. KEY WORDS: Time-domain;ultrasonicNDE; retrievefunction; thickness; wavespeed.

1. INTRODUCTION

wavespeed (c) of an elastic layer. A review of the extensive literature on this subject is beyond the scope of this work; a few pertinent papers are discussed in sequel. A widely used technique is the so-called pulse-echo method. (I-3) Here, the time interval between two successive echoes, &t, is measured and then either c or h is ascertained from the simple relation c = 2h/At. Another extensively used method is that of pulse interference. (4) Here, the repetition rate is adjusted to cause superposition of echoes; the repetition rate is measured very accurately and the thickness (or the velocity) is deduced. As the plate thickness decreases the time interval between two successive echoes decreases. Eventually, the echoes b e c o m e inseparable and the aforementioned methods cannot be used. For a typical, commercially available, highly-damped transducer the pulse duration is about 3X, where X is the wavelength. Therefore, these classical methods break down for plates with h < 2.5-3X. Accordingly, we define a plate to be " t h i n " if h < 3X. Now, for many applications h is of the order of 10 Ixm. Therefore, in order to use these methods the frequency would have to be larger than 150 MHz. The use of such high-frequency transducers and the as-

There are a number of situations of technological importance in which one wishes to carry out an ultrasonic NDE of the acoustic properties of a thin plate. A few examples are: adhesively-bonded joints; protective surface coatings (automobiles, aircraft structures, NASP, etc.); near-surface delaminations in composite materials; and free standing thin foils of metals and ceramics. Another strong motivation for developing a UNDE technique for thin plates is in the field of composite materials. Here, even if sufficiently high-frequency transducers are available, one prefers to use low-frequency transducers for the following reasons: at high frequencies the wave begins to " s e e " the microstructural details such as plies, resin-rich regions between plies, and individual fibers. The scattering phenomena interfere with the NDE process at the level of the plate thickness. Over the past quartercentury a wide variety of ultrasonic techniques have been reported for the measurement of the thickness (h) and Center for Mechanicsof Composites, Departmentof AerospaceEngineering, Texas A&M University, CollegeStation, Texas 77843.

121 0195-9298/93/0600-0121507.00/0 © 1993 Plenum Publishing Corporation

122

sociated electronics makes this approach cost-prohibitive. More recently, several frequency-domain techniques have been developed for NDE of thin plates. These may be broadly classified into three categories: resonance, (sI1) leaky-Lamb wave O2-23) and spectroscopy. (24-26) Recently, the authors reported a strictly time-domain technique for ultrasonic NDE of extremely thin plates, (z7,28) and so far as we know this is the first timedomain technique for plates whose thickness is about two orders of magnitude smaller than the wavelength, i.e. (h/X)= O(10-2). The transmitted (or reflected) field was expressed in terms of a sum of an infinite series involving the incident field. The series was found to converge rather slowly: about 30 terms for an aluminum plate in water and about 50 terms for a steel plate in water, for example. The plate properties were estimated through a comparison of the measured and calculated transmitted fields. The objective of this paper is to report a time-domain ultrasonic NDE technique which is computationally an order of magnitude more efficient than that reported in Refs. 27 and 28. Usually, the frequency-domain transfer function is defined as response/excitation.(29)In this work, we merely examine the inverse of the transfer function, i.e., the excitation/response, and call it the retrieve function. Whereas the inverse Fourier transform (IFT) of the plate transfer function was found to result in an infinite series; (27,2s) the IFT of the retrieve function consists of only two terms. Therefore, the incident field can be reconstructed from the transmitted field by a mere two-term summation. This is the main contribution of this paper. The new technique has been used for the measurement of the thickness and wavespeed of thin aluminum plates with thickness ranging from 0.089 to 6.426 mm using low-frequency, 1-MHz transducer, i.e., 0.01 < (h/ X) < 1.0. A systematical analysis of the sensitivity of the measurement procedure to thickness and wavespeed has been carried out. It is shown that a thorough and careful examination of the forward problem is a necessary prerequisite to the development of an inverse scheme. The measurement error for thickness and wavespeed was found to be about 1% and 3%, respectively. Finally, it appears that the new technique can be readily extended to timedomain NDE of layered media. To this end we will report only the theoretical results for two- and threelayered media (Appendix).

Zhu and Kinra

6 5

X

= 0

X = Cl+h

Fig. 1. Various reflections from and transmissions through a plate.

the incident wave traveling along the positive x direction (Ray 1 in Fig. 1), u t.... (x,t) = g ~ ( t - s ~ ) be the transmitted wave and uree(x,t) -- gr(t+so, ) be the reflected wave, where t is time, so is the slowness of water, and co is the wavespeed (So= 1/Co). The total transmitted and the reflected fields are (27) g(t

-

SoX) = oo

TolTlo 2 R ~ f [ t - SoX - (h(Zm + 1)s - So)]

(1)

m=O

g~(t + SoX) = -Raof[t + So(X - 2a)] + TmRloT m ~

Rz~m-1)f[t + So(X - 2a) -

2msh](2)

m=l

where s is the slowness of the plate, Tij is the transmission coefficient for a wave transmitted from medium i to j, T o.= 2Zi/(Zi + Z~), Rii is the reflection coefficient of a wave in medium i reflected fromj, R 0 = ( Z i - Zy(Zi + Zj), and Z-- pc is the acoustic impedance. For the case of an aluminum plate immersed in water,, the series (1) was found to converge in about 30 termsJ 27,2s) With "q= t - s d c , let F(eo) be the Fourier transform (FT) of f('q)

F(oJ) - ~

1 I

f(~l)e-i~nd'q

-~

< o~ < ®

(3)

where o~is the circular frequency. The associated inverse Fourier transform (IFT) is given by +~

1

2. THEORY -

Consider an infinite plate of thickness h as shown in Fig. 1 immersed in an elastic fluid (water) occupying the space a<__x<_b, b =a +h. Let u~n°(x,t) = f ( t - s ~ ) be

v2-4

f F(oj)eia~O&o

-co

< ~q < ~

(4)

_o

Similarly, let Gt(o~)be the FT ofg~('q). Following a fairly standard notation we define the transfer function of the

Ultrasonic N D E o f Thin Plates

plate in transmission

123

a s (29)

H*(o3) -

(5)

u(.,)

The transfer function is also known as the complex frequency response or the magnification factor. The transfer function for a single plate is well known (3°-a2) To1Tree-i.,(~-~o)h H*(to) = ? - - ~

(6)

We note in passing that H*(to) is independent of x and that the IFT of H*(oJ) is the infinite series (1) with f 0 replaced by the Dirac-delta function 80. We now introduce a retrieve function which is merely the inverse of the transfer function F( o)

Q t(o ) - Gt( )

(7)

F(o~) = Qt(m)Gt(o~)

(8)

1

(9)

Therefore

and -

[e i~o(so-s)h _ Rgle-,~o(s,,+~)h]

ro r o

least an order of magnitude saving in the computational effort to accomplish the same NDE objectives. All experimental work reported in this paper was done with the transmitted field. The thickness (or wavespeed) was estimated through a comparison of the measured and reconstructed incident fields. For completeness, however, we now treat the case of reflection. With ~ = t + s d c , let RmF(¢o) and Gr(eo) be, respectively, the FT of the front-face reflection, R o f ( ~ - 2soa ) (Ray 2 in Fig. 1), and the entire reflected field, g"(~), given by Eq. (2). We now introduce a transfer function of the plate in reflection as H*(o~)=G"(m)/Rof(O~), then(3O--32) e i2°~sh -- 1

H*(~o) - ei2O,,h _ Ro2

and is independent of x. Following the case of transmission, when we defined the retrieve function Qr(co)--- 1/ H*(m), it resulted in a q'(t) given by an infinite series which, of course, defeats the entire purpose of this paper. It is noticed in Eq.(2) that - R ~ o ( ( t + s o ( x - 2 a ) ) is the front face reflection (Ray 2 in Fig. 1) whose sign is opposite to the remaining pulses in the reflected field. Let Eq.(2) be rewritten as g~(~) - R m f ( ~ - 2soa) o~

Let q'('q) be the IFT of Q'(~o). It can be readily shown that (2,rr) I/2

=

(So - s)h]

ro,:qo

- RoZlS['q -

(So + s)h])

(10)

For later use, we define qx(t) = (2"rr)m (ToiTlo)-~g['q- (s 0 - s ) h ] and q2(/) = - ( 2 " r r ) 1/2 (TmTlo)-~Rm28[~l - (So +s)h]. Finally, given the transmitted field, the incident field can be calculated from Eq. (8) as a convolution of qt(~q) and od(~q) f(rl) = qt(~q)og'(.q) _

(13)

1

J

qt('r)gt(~q

-- "r)d'r

(11)

= rmRloTlo ~

R~Im-1)f(~ - 2Soa - 2rash)

The left hand side of Eq.(14) is the total reflected field minus the front face reflecton, i.e., Rays 6 + 10+ 14+ ...... 2. Taking Eq.(1) into consideration, Eq.(14) can be written as gr(~) - Rmf(~ - 2Soa) = R21ogt[~ - So(X + h + 2a)]

RmF(m) or

_ RolF(¢o)

1 -

H*(to)

-

1

(16)

or

1 f('q) - T m Tm ('g* ['q - (so - s)h] RZlgt[~l - (So + s)h])

(15)

If ~ ( { ) - R J ( ~ - 2 s o a ) is taken to be the total response one would expect to derive a relation similar to Eq.(12) for the case of reflection. This observation led us to introduce a slightly modified retrieve function Q~(m) = a r ( ¢ o )

-

(14)

m=l

1

Qr(fo)

(12)

We now compare Eqs. (1) and (12). In (1), an evaluation of g'(~q) from f('q) requires an infinite sum. In comparison, an evaluation of f("q) requires the summation of only two terms involving the g'(-q). This results in at

=

TolTlo(R21 -

e iz°'sh)

(17)

then q"(~) -

(2'rr)1/2 (R218({] 2sh)) TmTlo ~ " " - ~(~ +

(18)

124

Zhu and Kinra

Given the reflected field, the front-face reflection can be calculated from Eq.(16)

may be readily shown to be

RmF(to ) = Q"(eo)[G"(o~) - RmF(m)]

....

(2'rr) 1/2Rol (r~(~

(19)

then

- (So - s)h] - 6[~ - (so + s)h])

The attention is now turned to the retrieve function for reflection. Equation (18) may be rewritten as Rolf(~) = q({)o[g({) - Rmf({)]

(20) q~(~q) -

or

- (2"rr)1/2 ZolZlo

Rolf({) = Ro~f({ - 2sh) + g({) - RZ~g(~ - 2sh)

(8[(n

(21)

(So -

s)h + (So + s)h]

-RZlS['q - (So + s ) h + (So + s)h]) (23)

A comparison of Eq.(23)with Eq.(10) then gives the relation

Finally, we recall that the incident field is merely, uin¢(x,t) = f(t - SoX )

-

(22)

3. AN INTERESTING PHYSICAL INTERPRETATION OF THE RETRIEVE FUNCTION Let h(~) be the IFT of H*(eo)=G'(o~)/F(m). It is well known that h(qq) is the impulse response function or the Green's function and is the response to a Diracdelta excitation function. For the plate problem under consideration, it is an infinite sequel of equi-spaced Diracdelta functions of diminishing amplitude (see Eq.1). In view of Q'(eo) =F(o~)/G'(oJ), one would expect that q'(x)) is the excitation required to produce a Dirac-delta response; we will now show that this is indeed the case. The retrieve function, qt(.q), consists of two pulses: ql(t) and q2(t) (see Eq.10). Let's consider the first pulse ql(t) = (2"rr)m(ToxTlo)- lS['q - (So - s ) h ] normally incident along Ray 1. Upon through-transmission it produces Ray 4 given by (2"rr)l/28('q-soh). After suffering a pair of internal reflections a t x = a a n d x = b , Ray 7 is produced and is given by (2rr)mTlo-1Ro128['q- (So+s)h] whereupon it becomes clear that the second pulse, q2(t) (Ray 1' shown in Fig.l) of Eq.(10), coincides with Ray 7. Indeed, upon transmission at x = a Ray 1' becomes --(2v)l/2Tlo-aRm28[~q- (s O+s)h] which exactly cancels Ray 7. This is the main point of this discussion: once Ray 7 is canceled, there are no further internal reverberations in the plate and 8(rl -soh) comprises the entire transmitted field. Accordingly, q'(-q) may be called the impulse excitation function. We now calculate the reflected field due to qt(,q). Along Ray 2 it is (2rr)'d(TolT1o)-lRolS[~-(So-S)h]. Along Ray 6 there are two contributions: Ray 2' (reflection of Ray 1') and transmission of Ray 5. Their sum

qr(,q _ (So + s)h) = -q'("q)

(24)

Equation (24) simply tells that qr({) has exactly the same physical meaning as qt("q). Therefore, we conclude that q'(~l) and qr(~) are the impulse-excitations required to produce a transmitted field consisting of one Dirac-delta function and a reflected field consisting of two Dirac-delta functions, respectively.

4. EXPERIMENTAL PROCEDURES

A schematic diagram of the experimental apparatus is shown in Fig. 2. A pair of Panametrics broadband, water immersion, piezoelectric transducers with a center frequency of 1 MHz was used for generating and reI

I

Pulse Generator

]

1

='ioo G.... 'o.I

1

_____qP .... .o..,,,er t Water

Bath Receiver

Specimen

Digitizing

Signal

Analyzer

Fig. 2. Block diagram of the experimental setup.

Ultrasonic NDE of Thin Plates

125

ceiving the elastic waves. An experiment was initiated at time t = 0 by a triggering pulse produced by a Tektronix PG501 Pulse Generator. This pulse was used to trigger a Wavetek 190 Function Generator. The function generator produces a 1 p~s sinusoidal pulse which is amplified with an ENI A150 Power Amplifier to about 150 volts peak-to-peak amplitude and applied to the transducer. Simultaneously, the amplified signal was used to trigger the oscilloscope to minimize the system jitter. The received signal was sent to the oscilloscope and digitized at a sampling interval, At_<10 ns; (27) At--1,2 and 4 ns were used in this work. A NEC 486 computer was used to control all the operations of the digital oscilloscope through an IEEE 488 bus. The digitized information was sent to the computer for further analysis.

The essence of our inverse scheme is as follows: the best estimate o f p is the one which minimizes the difference between f(t;p) and ft(t), i.e., minimizes Es(P).

6. S E N S I T M T Y ANALYSIS In order to quantify the sensitivity off(t;p) top, we introduce a time-averaged sensitivity

T

where fma~--MaXf(t)l and SI,p is the (time-averaged) normalized change in f(t;p) caused by a normalized change in p. In the present case of discrete data,

5. NUMERICAL PROCEDURE

Si,p =

If a closed-form analytical expression for g'(t) was known then Eq.(12) would be sufficient for calculating the incident field. However, the waveform received by a typical transducer cannot be easily described in terms of elementary functions. Instead g(t) is acquired through the process of digitization; let At be the sampling interval. Then g(t) is known only to the extent of a set of number pairs

fmaxl

g'(t):{gJ, tl}

j = 0, 1, 2 . . . N

with tj = j At and the total time interval T = NAt. Note that in order to calcuIate f(t) at an arbitrary time, one must know g(t) for all values of its argument. But g(t) is known only at discrete times. Therefore, the Lagrange three-point interpolation method (33) was used to construct a continuous g(t) from the discrete {g/,tj}. Letf(t;p) be the reconstructed incident wave; where p =h or c is the parameter to be determined. Let f*(t) be the measured incident wave. A root-mean-square error function is introduced to quantify the difference between the theory and the experiment,

1 (lf O

EI(P) = f"m,~-----~

)1

(f(t; p) -- f*(t))edt ~

i=.

(f(t, ;p + Ap)

f(t,

)

For a perfect measurement, E/~p) depends only on Ap, i.e., Es(Praun)= 0. In reality, E)(p) depends on both the deviation, Ap, and random measurement error, Aft(t), i.e., even when P=PTRuE, E I ( p ) > 0 - M o r e o v e r , Aft(t)=f(t;PwRun)--ff(t). An approximate connection between Ap and Aft(t) may be obtained as follows. Let Ap be the deviation required to produce the root-meansquare error, Ei(p)--Ei(Af*), where Ef(Af*) is the normalized time averaged random measurement error. In Eq.(25), by using a Taylor series expansion of f(t;p) aboUtpTRUE (i.e.,p =PTRuE + Ap) and retaining only the term linear in Ap, it can be shown that

T Ef(p) = EI(Af* ) =

-T

T

(1!

\ Op Ap

i

(f(ti;p) - f*(ti)) 2 ~

;)

1/2

dt

. \ fmax 01)]

]

PTRUE

Taking Eq.(27) into account, Eq.(29) can be written as

zip = Ei(Af*) prau E

= -f~,x

(28)

(25)

where T is the total duration of the pulse, f*(t). For the present case of discrete data, Eq.(25) takes on the form

E~)

W

(26)

where f~,x = MAXf*(t)I, N is the total number of points at which f~'(t) is measured, ft(ti) is the measured signal andf(tdp ) is the reconstructed quantity computed at t = t i.

(30)

Ji, p

Equation (30) may be used to estimate the measurement error. The main point of this discussion is that when the sensitivity, St,p, is small, a small measurement error, Aft(t, would result in large error, Ap, and vice versa. Equation (28) was used in calculating the sensitivity, Si,p, arbitrarily taking (Ap/pTRuE) to be 10 -z in all the computations. The results are shown in Figs. 3 and 4.

126

Zhu and Kinra h/X 10.0

r.~.

0.0 0.2 0.4 :~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.6

0.8

1.0

, ..........

8.0

6.0 =.

uT 4.0

0.0

~,,,,, 0.0

2.0

,,~,,,,,I 4.0

, 6.0

velocity will result in changes in T12T2I , R21 and the round trip time, t,.= 2h/c = 2sh, inside the plate. A change in T12Tzl and R21 will change the magnitude of the rays shown in Fig.1 while a change in tr will introduce a time shift of these rays. It can be shown that as c increases, the derivatives with respective to c of T12T2~, R212 and t, decreases. Therefore the sensitivity of SI,, generally decrease as c increases. Note that the curves shown in Figs. 3 and 4 are specific to f(t) used in this research which is shown in Figs. 7-9. The argument concerning the behavior of Si,p vs. p is universal, however. The sensitivity of the reconstructed signal, f(t), to a _+5% change in h and c is shown in Figs. 5 and 6 (_+5% is arbitrarily chosen as an acceptable error in deducing h or c). Since wavespeed in aluminum is larger

h(mm) F i g . 3. Sensitivity, Si,,, (

) and SI,~ ( . . . . . . ) as a function of h for an aluminum plate.

0.60

0.40

6.0

- ..... ....

h=2 mm 1.05h 0.95h

9.0

11.0

O

>

0.20

S

5.0

uJ

0.00

£:3

4.0

-0.20 o_

3.0

<

-0.40 2.0 -0.60 5.0

1.0

7.0

TIME (/~s)

0.0

rlllllll]lllrllrlrl

0.0

2.0

........

4.0

~1 . . . . .

6.0

IIiIl,lrlllll

8.0

10.0

Fig. 5. Effect of 5% change in thickness, h, on the reconstructed signal, f(t).

(m~/#,) Fig. 4. Sensitivity, SI.~, as a function of the wavespeed, c(h=2

ram).

0.60 --..... ....

0.40

It is noted in Fig. 3 that both Ss,h and S~c increase with h. This plot quantifies what is intuitively obvious; it is easier to carry out NDE of thicker plates. Conversely, one cannot carry out NDE of a vanishing thin plate, Even if one could make an extremely accurate measurement ofF(t), i.e., solve the Forward Problem very accurately, one cannot solve the Inverse Problem. Generally, S¢;,h > Szc, therefore, we know a priori that we can expect a more accurate measurement of thickness than wavespeed. The sensitivity to wavespeed, Sy,c, is plotted against c in Fig.4, where c is treated as a variable. It is noted that Si,. decreases very fast as c increases from zero, This may be explained as follows. A change in the wave

c=6.35 1.05c 0.95c

mm/czs

...I O

>

0.20

S -0.00 £13

_q - 0 . 2 0 13_ ]Z <

-0.40

h

~0.60

iiirll~rllll~llllll~lllllrrl

5,0

7.0

=

2

mm

Iiir~

9.0

11,0

TIME (/~s)

Fig. 6. Effect of 5% change in wavespeed, c, on the reconstructed signal, f(t).

Ultrasonic NDE of Thin Plates

127

than that in water, when h is increased f(t) shifts to the right; it also results in a decrease in the peak amplitude. The effect of an increase (a decrease) in wave speed is similar to that of a decrease (an increase) in plate thickness. With reference to the amplitude resolution of the oscilloscope used (10 -3 in the normalized magnitude), this change in f(t) is an order of magnitude larger than the measurement error, it is clear that both h and c can be estimated with an accuracy better than 5%.

0.28

o >

0.08

I

f

~2 2 -0.12

7. RESULTS AND DISCUSSION

- 0 . 3 2 ~......... ' ......... " 165.0 166.0 167.0

All NDE proceeds in 2 stages: (1) The forward problem: given the plate thickness and wavespeed, compare the measured signal, f*(t), and the theoretically predicted (reconstructed) signal, f(t;p); (2) The inverse problem: given f*(t) and g(t) deduce either h or c from a comparison off(t;p) and f*(t).

TIME

168.0

169.0

170.0

(/zs)

Fig. 7. Comparison of the reconstructed signal, fit), and the measured signal, f*(t), (he = 0.089 mm, h/X = 0.014, fit) ...... .f*(t)).

0.28

7.1. The Forward Problem Eight aluminum plates with thickness ranging from 0.089 mm to 6.426 mm were tested with 1-MHz transducers (see Table I). For three typical values of h, namely, h = 0.089, 0.254 and 1.003 ram, f(t) and f*(t) are compared in Figs. 7, 8, and 9, respectively. The comparison is considered excellent. In all cases, Ei was found to be less than 4%.

Given c, Find h. Here we assume the wavespeed is known, c = 6.35 mm/txs, and we use the NUMERICAL PROCEDURE to deduce the thickness, h. Accordingly, the normalized error Ej(h) (Eq.26), is plotted against h Table I. Thickness and Wavespeed Measurement Data"

Thickness

Wavespeed CNDE

Error Error(mm/ Error Error h~(+-3"10 -3 mm) hc/X hNnz (ram) 0xm) (%) ixs) (mm/p.s) (%) 0.014 0.04 0.082 0.158 0.234 0.358 0.5 1.01

" CTRUE = 6.35 mm/g,s.

0.089 0.255 0.526 1.000 1.476 2.259 3.169 6.35

0.08

S if3

J -0.12 Q-

ill1 .... ~ ........ illlrllllllll~llIllll,l~lrlll~ -0.32 165.0 166.0 167.0 168.0 169.0

7.2. The Inverse Problem

0.089 0.254 0.521 1.003 1.486 2.276 3.175 6.426

>

0.0 1 5 3 10 17 6 76

0.0 0.4 1.0 0.3 0.7 0.7 0.2 1.2

6.70 6.15 6.20 6.25 6.22 6.23

0.35 0.2 0.15 0.1 0.13 0.12

5.5 3.0 2.3 1.6 2.0 1.9

TIME

170.0

(/zs)

Fig. 8. Comparison of the reconstructed signal, fit), and the measured signal, f*(t), (he = 0.254 mm, h/k = 0.040, fit) ...... .f*(t)).

in Fig.10 and the corresponding measured if(t) is shown in Fig. 8. The plate thickness was measured with a pair of calipers and was found to be hc=0.254 mm +-_2.54 Ixm. The numerical search is carried out over 0.0
128

Zhu and Kinra h/he 0.26

1.00

0.62 0.82 1.02 1.22 • .....................................

1.42

4.0

9.0

>

0.80

0.06

tm

0.60

J -0.14 2~ <

o bA

-0.34 166.0

0.40

0.20 167.0

168.0

169.0

TIME

170.0

171.0

(,u,s) 0.00

Fig. 9. Comparison of the reconstructed signal, f(t), and the measured signal, f*(t), (h~= 1.003 ram, h / X = 0 . 1 5 8 , f(t) ....... f*(t)).

5.0

6.0

7.0

h

8.0

(ram)

Fig. 11. Plot of error function, E/h) (he =

6.426

mm, h/X = 1.0).

h/he 0.0 0.60

0.50 1.00 ..........................

1.50 2.00 , .........

0.70

0.50 d O

>

3 0.20

0.40

u7 b7

0.50

fD

L

-J - 0 . 3 0

0.20

<

0.10 h0 -0,80

i111r

170.0

0.00

0.0

0.1

0.2

0.3 h

Fig. 10. Plot of error function,

0.4

0.5

llr

iii

i~

iii

175.0 TIME

i i l 4 1 1 r

180.0

r fir

i ,__

185.0

(/~s)

(mrn)

E/h) (h~=

0.254 mm,

h/X =

0.04).

and the location of the minimum becomes proportionally imprecise. Figure 10 typifies all sub-half-wavelength plates, (h/X) < 0.5. Whenever (h/X) > 0.5, an additional problem arises and this is discussed next. Figure 11 shows the result of a numerical search over a large range of h: 4 < h < 9 mm or 0.62
Fig. 12. Comparison of the reconstructed signal, fit), and the measured signal, if(t) ( f(t) (h = 6.35 m m ) . . . . . . . f*(t)).

the corresponding EXTREMA o f F ( t ) and we obtain a true minimum of Ec(h); this situation is shown in Fig.12. Since wavespeed in aluminum is larger than that in water, when h increases the peaks off(t;h) shifts to the right. Eventually, the first peak off(t;h) aligns with the second peak o f F ( t ) (Fig.13) and we get a spurious minimum to the right of the true minimum (Fig.ll). Conversely, as h is decreased from its true value, peaks off(t) move to the left, the second peak off(t;h) aligns with the first peak o f F ( t ) and we get another spurious minimum to the left of the true minimum. However, it is noted in Figs.12 and 13 that as h deviates from its true value large fluctuation occurs in the trailing part off(t;h). Such poor comparison (large E/(h)) at these spurious minima

Ultrasonic NDE of Thin Plates

129 o/O,.o~

0.70 0.62

0.82

1.02

1.22

1.42

1.00

g 0.20

0.80

S "6"

D

"~

0_ -0.30

0.60

c o L,J

-o.8o

......................... 170.0

175.0

I 180.0

TiME

0.40

0.20

185.0 c

(/J.s)

0.00

Fig. 13. Comparisonof the reconstructedsignal,fit), and the measured signal, if(t) ( fit) (h =4.5 mm), -..... fit) (h =8.1 ram).......

f*(t)).

4.0

5.0

6.0

o Fig. 15. Plot of error function,

C/CTRUE

0.16

0.62 0.82 1.02 1.22 ......... , ......... , ......... , .......

1.42

0.12

LJ

o

0.08

Ld 0.04

0.00

4.0

5.0

6.0

7.0

8.0

9.0

o (mm/ffs) Fig. 14. Plot of error function,

E/c)

(he= 1.003 ram, h/X=0.158).

7.0

8.0

9.0

(m~/,~s)

E/c)

(hc=6.426 mm,

h/k= 1.0).

the numerical search range only the true minimum can be seen, which was found to be at c = 6 . 2 3 mm/p~s. This concludes the development of the inverse algorithm. Eight aluminum plates were tested and the inverse algorithm was applied to them. The thickness and wavespeed measurement results are presented in Table I. The measurement error is defined as e = ~ONDE - p T R U E I / PTRUE" The thickness measured by a pair of calipers, he, was taken to be our " t r u e " value while 6.35 mm/~s (a4/ was taken as the " t r u e " wavespeed. The typical error in thickness measurement is of the order of 1%. When plate thickness is larger than 1 mm, the typical measurement error in wavespeed is less than 3%. The smaller error in thickness measurement than that in wavespeed measurement is consistent with the fact that sensitivity Si, h is larger than Si, c. The absence of measured wavespeeds for h = 0.089 and 0.254 mm in Table I is due to the small sensitivity, Si,c at these two thickness/wavelength ratios. The tabulated data is plotted in Figs.16 and 17.

guarantees the global minimum is always near the true thickness. The numerical search gives a best estimate of h = 6.35 ram.

8. S U M M A R Y

Given h, Find c. Here we assume the thickness is known and we use the NUMERICAL PROCEDURE to deduce the wavespeed. Accordingly, Fig.14 shows Ei(c ) as a function of c for a 1.003 mm thick plate. The corresponding measured if(t) is shown in Fig.9. The numerical search carried out over 4 < c <9 mm/p,s gives a best estimate of CNDE-----6.15 mm/txS. The error function Ej(c) as a function of c for the 6.426 mm thick plate is shown in Fig.15; El(c ) could have three minima. However, within

A new NDE technique has been developed to determine the properties of a thin plate using only timedomain information. A retrieve function for the reconstruction of the incident field for both cases of through transmission and reflection was derived. A sensitivity analysis of this technique to the parameters to be determined has been carried out. This technique has been applied to measure the thickness (0.089_
130

Zhu and Kinra hc/X 0.0

0.2

0.4

0.6

0.8

1.0

University, College Station, Texas. Thanks are also extended to Mr. P. Jaminet for his helpful comments.

REFERENCES '7

4

-E

2

0

~ r , r , l l l

0

lip

2

. . . . .

,

, , , , , i r l l l l

4 hc (mm)

6

Fig. 16. Plot of hNDE VS. hc (0.014
h/X 0.0.

0.2

0.4

0.6

0.8

1 .O

CTRUE= 6.35 mm//~s 8

.-2 E E

7

w

6

5

2

4

5

h(mm) Fig. 17. Plot of CNDEVS. hc (0.08-h/X-
and the wavespeed (1.003
ACKNOWLEDGMENT

The material is based on the work supported by the Texas Advanced Research Program (Advanced Technology Program) under Grant No. 282 to Texas A&M

1. E. P. Papadakis, Ultrasonic velocity and attenuation: Measurement methods with scientific and industrial applications, in Physical Acoustics Principles and Methods (Vol.12), W. P. Mason and R. N. Thurston, eds. (Academic Press, New York, 1976), pp.277-374. 2. N. P. Cedrone and D. R. Curran, Electronic pulse methods for measuring the velocity of sound in liquids and solids, J. Acoust. Soc. Am. 26:963-966 (1954). 3. G. M. Light, G. P. Singh, and F. D. McDaniel, Ultrasonic and X-ray fluorescence measurement of the thickness of metal foils, J. Mater. Eval. 47:322-330 (1989). 4. H. L. Mcskimin, Pulse superposition method for measuring the velocity of sound in solids, J. Acoust. Soc. Am. 33:12-16 (1961). 5. D. I. Bolef and M. Menes, Measurement of elastic constants of RbBr, Rbi, CsBr and CsI by an ultrasonic C. W. resonance technique, J. Appl. Phys. 81:1010-1017 (1960)o 6. E. A. Lloyd, Non-destructive testing of bonded joints, NDE Int. 7:331-334 (1974). 7. J. L. Rose and P. Meyer, Ultrasonic signal-processing concepts for measuring the thickness of thin layers, J. MaWr. Eval. 32:249255 (1974). 8. F. H. Chang, J. C. Couchman, and B. G. W. Yee, Ultrasonic resonance measurements of sound velocity in thin composite laminates, J. Composite Mater. 8:356-363 (1974). 9. G. Alers, P. Flynn, and M. J. Buckley, Ultrasonic techniques for measuring the strength of adhesive bonds, J Mater. EvaL 35(4):7784 (1977). 10. C. H. Yew and X. W. Weng, Using ultrasonic SH waves to estimate the quality of adhesive bonds in plate structures, J. Acoust. Soe. Am. 77:1813-1823 (1985). 11. A. Sinclair, P. A. Dickstein, J. K. Spelt, E. Segal, and Y. Segal, Acoustic resonance methods for measuring dynamic elastic modulus of adhesive bonds, in Dynamic Elasticity Modulus Measurements, ASTM STP 1045, A. Wolfenden, ed. (American Society for Testing and Materials, Philadelphia, (1990), pp.162-179. 12. C. C. H. Guyott and P. Cawley, Evaluation of the cohesive properties of adhesive joints using ultrasonic spectroscopy, NDT Int. 46:233-240 (1988). 13. R. D. Adams and J. Coppendale, Measurement of the elastic moduli of structural adhesives by a resonant bar technique, J. Mechanical Eng. Sci. 18:149-158 (1976). 14. E. Henneke, Reflection-refraction of a stress wave at a plane boundary between anisotropic media, J. Acoust. Soc. Am. 51:210217 (1972). 15. Y. Bar-Cohen and D. E. Chimenti, Nondestructive Evaluation of Composite Laminates by Leaky Lamb Waves, Douglas Paper 7598, McDonnell Douglas Corp., Long Beach, California (1985). 16. M. de Billy and L. Adler, Measurements of backscattered leaky lamb waves in plates, J. Acoust. Soc. Am. 75:998-1000 (1984). 17. P. B. Nagy, A. Jungman, and L. Adler, Measurements of backscattered leaky lamb waves in composite plates, J. Mater. Eval. 46:97-100 (1988). 18. D. E. Chimenti and A. H. Nayfeh, Leaky lamb waves in fibrous composite laminates, J. AppL Phys. 58:4531--4538 (1985). 19. V. Dayal and V. K. Kinra, Leaky lamb waves in an anisotropic plate. I: An exact solution and experiments, J. Acoust. Soc. Am. 85:2265-2276 (1989). 20. V. Dayal and V. K. Kinra, Leaky Lamb Waves in an Anisotropic Plate. II: NDE of Matrix Cracks in Fiber Reinforced Composites, Aerospace Engineering Dept. Report, 1989, Texas A&M University, College Station, Texas 77843.

Ultrasonic NDE of Thin Plates 21. S. I. Rokhlin and W. Wang, Critical angle measurement of elastic constants in composite material, Y. Acoust. Soc. Am. 86:18761882 (1989). 22. A. K. Mal, Guided waves in layered solids with interface zones, Int. J. Eng. Sci. 26:873-881 (1988). 23. M. R. Karim, A. K. Mal, and Y. Bar-Cohen, Inversion of leaky lamb wave data by simplex algorithm, J. Acoust. Soc. Am. 482491 (1989). 24. V. K. Kinra and V. Dayal, A new technique for ultrasonic nondestructive evaluation of thin specimens, J. Exp. Mech. 28:288297 (1988). 25. V. K. Kinra and V. Iyer, On the use of phase spectra for ultrasonic NDE, in Proc. of the 1990 SEM Spring Conference on Experimental Mechanics (Albuquerque, New Mexico, 1990), pp.478486. 26. V. Iyer and V. K. Kinra, Frequency-domain measurement of the thickness of a sub-half-wavelength adhesive layer, in Proc. of the 1991 SEM Spring Conference on Experimental Mechanics (Milwaukee, Wisconsin, 1991), pp.668-675. 27. C. Zhu and V. K. Kinra, Time-domain ultrasonic measurement of the thickness of a sub-half-wavelength elastic layer, J. Testing Eval. 20:265-274 (1992). 28. V. K. Kinra and C. Zhu, Time-domain ultrasonic NDE of the wave velocity of a thin layer, J. Testing EvaL 21:29-35 (1993). 29. L. Meirovitch, Analytical Methods in Vibrations (The Macmillan Company, New York, 1967). 30. L. M. Brekhovskikh, Waves in Layered Media (Academic Press, San Diego, California, 1980). 31. S. E. Hanneman and V. K. Kinra, A new technique for ultrasonic nondestructive evaluation of adhesively-bonded joints, Part I. Theory, J. EXp. Mech. 32:323-331 (1992). 32. S. E. Hanneman, V. K. Kinra, and C. Zhu, A new technique for ultrasonic nondestructive evaluation of adhesively-bonded joints, Part II. Experiment, J. Exp. Mech. 32:332-339 (1992). 33. J. G. Herriot, Methods of Mathematical Analysis and Computation (John Wiley & Sons, New York, 1963).

131 1

Awl) -

(gt(,q --

ToI

+ RmR12gt(~q

t0 + L1 + /-2)

-

to -

t 1 + t2)

+ R12R20gt('q - t 0 + tl - t2) + R o l R z o g t ( ' q - to - tl - t2)) to = s o ( h i + h2)

i = 1, 2.

ti = sihi,

A2. The relationship between the incident and reflected fields for a 2-layered medium: gr({)

= Rolf(~) + R 1 2 ( f ( { - 2t~) - Rolg"({ ,7 2tl)) + R12R23(Rmf(~ 2t2) - g ( ~ - 2ta) )

+ R 2 3 ( f [ ~ ' - 2(t 1 + t2) ] - R o l g r [ ~ - 2(tl +

t2)])

ti = s i h i ,

i = 1, 2.

A3. The relationship between the incident and transmitted fields for a 3-layered medium: f(~) =

1 To1T, eTe3T3o(gt(~a

to + tl + t2 "OV t3)

-

tl + t2 + t3)

+ RolRlzgt(~q

-

to -

+ RlzR23gt('q

-

to + tl

+ R23R30gt('q + Ro~Rz3gt("q

-

to + tl + t2 - t3) to - tl - t2 + t3)

+ RlzR30gt(~

-

t o + t~ -

t2 -

+ RoxR3ogt("q

-

to -

t2 -

+ RolRlzRz3R30gt('q

tl -

to = so(hi + h2 + h3)

-

t2 +

to -

t3)

t3) t3) t3) ) i = 1, 2, 3.

tI + t2 -

ti = s i h i ,

A4. The relationship between the incident and reflected fields for a 3-layered medium: APPENDIX

gr(~)

_~ R m f ( ~ )

+ R~2(f( ~ _

+ R12R23(Rmf(~

Consider an N-layered (N = 2, 3) medium immersed in water occupying the space a < x < a + E N = l h i , where hs is the thickness of the ith layer. Let s~ be the slowness of the ith layer, and R o and TU as defined before. The relationship between the incident and transmitted (reflected) fields for two- and three-layered (twolayered) medium are given below.

2t2) - g~(~ - 2t2))

-

2t3)

+ Rz3R3o(Rmf(~ +

z(t

+

Rmg"[~

-

-

-

2t,) - Ro,g"(~ - 2t~)) g~(~

2t3))

2(6i + tz)])

- 2(~ z + t3)] - g,'[~ - 2(t 2 + t3)])

+ R12R30(Rolf[~

+ R12R23R3o(f[~ -

2(q + t3) ] - 2(tl + t3)])

-

Rmgr[~

+ R3o(f[~ - 2(q + t 2 + t3)]

A1. The relationship between the incident and transmitted fields for a 2-layered medium:

-

Rmgr[~

-

2(tl + t2 + t3)]) ti = s i h l ,

i = 1, 2 , 3.