Ultrasonic measurements for the prediction of mechanical strength

Ultrasonic measurements for the prediction of mechanical strength B.R. TITTMANN

Results are presented on the scattering of ultrasonic waves from defects embedded in metals and on the influence of these defects on the mechanical properties. A simple correlation has been discovered between an empirical expression for the ductility of a sample containing a single void and the back-scattered ultrasonic'power derived from scattering theory. An expression is derived which relates the ductility (reduction in cross-sectional area at tensile fracture) with the high-frequency (15 MHz) backscattered ultrasonic signal in an approximately linear fashion. The data obtained on diffusion bonded Ti-alloy samples are in good agreement with the predictions. These results open the door to the prediction of ductility from ultrasonic measurements.

The ultimate objective of most non-destructive evaluation, nde, studies is to develop a capability of pre-determining the in-service failure probabilities of a structural component, with the best possible confidence. Consequently, there has been considerable interest in establishing correlations between ultrasonic measurements and mechanical properties. For example, Regalbuto I established a relationship between the ultimate tensile strength of low strength bond samples prepared by the hot retort-cold press method and ultrasonic pulse echo measurements. More recently, Vary 2 developed empirical evidence to show a relationship between fracture toughness, yield strength and ultrasonic attenuation properties of metallic materials. We have presented work on the scattering of ultrasonic waves 3 from geometric obstacles embedded in metals and on the influence of these defects on the mechanical properties. 4 In this paper the discovery is reported of a correlation between the ductility (reduction of cross-sectional area at fracture) measured on tensile specimens and the back-scattered ultrasonic power from a single void embedded in the same specimens. A model is proposed and a simple relationship derived which predicts the ductility from ultrasonic measurements. Data obtained on cavities in Ti-alloy samples are in good agreement with the relationship. Since at this time there are no detailed theoretical treatments of the fracture process, this result should be useful to the field ofnde and fracture mechanics and should also provide an interesting physical link between the purely elastic process of ultrasonic wave propagation and the highly inelastic process of plastic deformation. The correlation is based on the realization that in a tensile specimen with a macroscopic cavity, both the power in a back-scattered ultrasonic wave and the ductility at fracture depend on the 'size' (that is the ultrasonic scattering crossScience Center, Rockwell International, Thousand Oaks, California

91360

NDT INTERNATIONAL. FEBRUARY 1978

section of the void) of the cavity. More specifically, in the case of the ultrasonic measurements, an increase in size, ie an increase in the ultrasonic scattering cross-section of the void, raises the power of the back-scattered wave as predicted by scattering theory. In the case of ductility an increase in size refers to an increase in the void's fractional volume (as described in an empirical relation) which reduces the ductility in a tensile test. While, in general, the relationships for both of these quantities are complicated functions of many variables, in certain regimes they become sufficiently simple to allow a correlation between them. The following sections present, after some discussions of sample preparation and measurement techniques, the model and its predictions together with experimental results supporting the model.

Sample preparation Basic to the technique of sample preparation is the diffusion bonding processs in which two metal surfaces are bonded in a way that removes all traces of the bond line. By way of example, Fig. 1 shows a cross-section of a hemispherical void created by diffusion bonding together the end faces of two short cylinders, one with a smooth surface and the other with a centrally located, ground-in hemisphere. A detailed look at the region where the bond was made shows the complete disappearance of the bond line by grain growth across it. Ultrasonic non-destructive measurements carried out in the frequency range 2.5 MHz to 15 MI-Iz also do not reveal the presence of the bond line. For the measurements discussed here, commercial grade Ti-6% AI-4%V was machined into the form of right circular cylinders, each about 20 mm in diameter and 30 mm long. The actual specimen was then prepared by taking a pair of these cylinders and diffusion bonding their end faces

17

commercial broad band transducer can effectively average them out. In this regime for a scatterer in the shape of an ellipsoid of revolution, the power of the back-reflected wave Sbs is to a good approximation equal to one-fourth the product of the acoustic impedance coefficient R and the two principal radii of curvature a i and a2 of the surface perpendicular to the direction of propagation of the incident wave, 6 or

Sbs

(1)

= (1/4)ala2R

Here R takes into account the acoustic impedance of the scatterer relative to that of the propagation medium and is equal to unity for a void. The power Prc (in units of dB) actually received at the terminals of the transducers (used in the pulse-echo mode) is Prc = 10 l o g l o C l S b s

(2)

where cl takes into account all the losses due to attenuation ,~ in the medium, beam spreading, bond absorption, and transducer efficiency. An accurate determination of c1 is difficult but has been accomplished in special cases. 7 In this work the measurement procedure discussed in the next section takes into account most of these losses by a normalization technique. The remaining losses c are determined by adjusting c to obtain a good fit between the experimental data and the calculations from the model. Combining Equations (1) and (2) and specializing the problem for a spherical void witha~ = a2 = a and R = 1 gives p = 10 logloca 2 or

(3)

p = B In ca 2

Fig. 1 Micrograph of the cross-section of a hemispherical cavity produced by diffusion bonding t w o machined sections of Ti-alloy; the lower figure shows an enlargement of the section where the bond was made and demonstrates the complete disappearance of the bond line by grain growth across it; the top figure is a mosaic of several micrographs

where the reflection coefficient p represents the normalized power received at the terminals of the transducer, B = 10 logwe, and c is a constant (including the factor 1/4) to be determined in fitting the (scattering) theory to the data.

0 I

together to form one cylinder 60 mm long. Prior to bonding, the surfaces to be mated were machined in such a way that when mated, the desired shape and location of the void would result on the bond plane. In the diffusion bonding process the mating surfaces were pressed together under a uniaxial pressure of 3.45 MN/m 2 for 30 minutes at 1100 K in a vacuum of 1.3 10 -4 N/m 2. Once bonded, the resulting specimen was machine finished into a cylinder with smooth sides and polished end faces.

Ultrasonic scattering from voids For ultrasonic reflection the regime of interest lies in the high frequency domain where the radius, a, of the cavity is small compared to the wavelength, X. As calculated from scattering theory ~ and shown in Fig. 2, the power backreflected from a spherical void rises dramatically at low frequencies, but near frequencies corresponding to X = 2~ra begins to oscillate about an approximately constant level. At high frequencies (X "~ a) the oscillations are seen to have a sufficiently small amplitude, so that a typical

18

Wavelength (mm) 1.0 0.5 I ]

2.0 1

0.35 I

-I0 -20

2A3 = I. 2

o -40 o. -50 -60 -70 I 2

I 4

l 6

I 8

I I0

I 12

L 14

I 16

1 18

20

Frequency (MHz) Fig. 2 Frequency (wavelength) dependence of power back-reflected from a spherical void embedded in Ti-alloy as calculated f r o m theory; ~ the t w o curves marked 2A3 = 1.2 and 2A1 = 0.4 correspond to diameters 1.2 mm and 0.4 mm respectively

NDT I N T E R N A T I O N A L .

F E B R U A R Y 1978

the line before the receiver input of the Immerscope. With the amplitude of the received signal aligned with an arbitrarily chosen reference mark on the oscilloscope the attenuator reading was recorded for the reference sample and the unknown sample. The two attenuator readings represent the amplitudes of the reflections from the unknown and reference samples expressed in units of power (dB). Since these units involve logarithms, a subtraction of the two attenimtor readings is equivalent to a division by the reference amplitude. This difference in the attenuator readings is the value referred to as the reflection coefficient throughout the text and on the graphs. The motivation for dividing by a reference amplitude as described above is to divide out as nearly as possible the frequency dependence of the transducer efficiency, transmitter power, and ultrasonic attenuation in the Ti-alloy.

-I0

IO00vm diem round end of hemisphere

-2O

~

~"

"O v

"E *~ -3 0

-- ~

IO00pan diem

500+

k~

a

n

u sp ere

~

g -4o f

500/~m diofi~ round end of hemisphere

o~ - 5 0

-60

-70

'~.

I

0

I I0,0 Frequency (MHz)

5.0

Fig. 3 shows representative data obtained at high frequencies (small wavelenths) for a few shapes and sizes of voids as measured on diffusion bonded Ti-alloy samples. At first glance the features displayed in Figs 2 and 3 appear very dissimilar. However, it must be emphasized that the calculations in Fig. 2 are for monochromatic waves, whereas the data in Fig. 3 are for broad-band transducers. Note that at 15 MI-Iz the difference in the reflection coefficients between the 1000/am and 500/am diameter spheres is approximately 8 dB, in reasonable agreement with the predicted 6 dB from Equation (3) - and that if both curves were shifted an equal amount (~ 10 dB), their position would be in reasonable agreement with those shown in Fig. 2.

I

15.0

Fig. 3 Plot of reflection coefficient for spherical and hemispherical cavities with 500 pm and 1000 #m diameter as obtained in ultrasonic measurements on diffusion bonded Ti-alloy samples; the reflections are normalized with respect to a reference scatterer, a flat plane air/ metal interface

The ultrasonic measurements were carried out with the single-ended pulse-echo mode so that for each frequency only one transducer was used for transmitting the waves and for receiving the scattered wave directly back from the obstacle. A modified Immerscope was used with a set of Automation Industries 13 mm transducers operating at set frequencies of 5, 10, and 15 MI-Iz. Only longitudinal mode transducers were used together with water bonds so that effectively little or no shear wave energy could be detected. In the measurement, the amplitude of the reflections from the various obstacles was divided by the amplitude of the reflection from a reference obstacle, chosen to be a free surface (metal/air interface) at the same location as the obstacle; ie half the length of the usual test specimen. This was done with the aid of a precision attenuator inserted in

D u c t i l i t y at t e n s i l e f r a c t u r e

For the tensile tests, the samples discussed under the heading 'Sample preparation', were machined into tensile specimens with a diameter of 6.35 mm and with the bond plane lying across the diameter of the centre of the gauge length. They were then pulled apart in a conventional tensile test apparatus at a strain rate of 4 x 10 -4 per second. Table 1 lists the specimens prepared, the measured yield stress, the ultimate tensile strength and the ductility (reduction of croa-sectional area).

Table 1. Tensile data for Ti-alloy samples with spherical and hemispherical defects of different sizes

Defect Specimen Shape

1 2

3

Material

Diameter (/am)

Cavity

Sphere .

.

.

.

Hemisphere

Cavity

0.2% offset yield (MN/m 2)

ultimate strength (MN/m 2)

Ductility Ao e = In -Af

No

Configuration and spacing

1000

1

-

773

863

0.137

500

1

-

773

871

0.282

500

1

-

782

876

0.323

4

. . . .

500

6

Array 1000 #m

794

885

0.155

5

. . . .

200

10

Array 500/am

794

884

0.200

6

. . . .

400

32

Mat. 1000/am

794

871

0.124

7

. . . .

200

96

Mat. 500/am

776

871

0.124

801

895

0.358

17

No defect

.

.

.

.

NDT INTERNATIONAL. FEBRUARY 1978

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from the coalescence of growing holes, but rather the initiation of a shear crack at a side of a hole marked the critical stage in the occurrence of fracture. Metallographic microstructure examination was used to measure the plastic strain distribution as close as 0.01 mm from a hole, and the change of the curvature of a hole was related to the radius of a hole. From these findings, a criterion of the onset of a shear crack was proposed. The criterion was derived from the comparison of plastic strain energy associated with the growth of a hole with that associated with the initiation of a shear crack. When it was applied to the case of metals with a uniform distribution of spherical cavities, 1°'11 the predicted decrease in the ductility by the increase in the volume fraction of inclusions was in good agreement with the experiment. The similarity between Nagumo's predictions and our data for the dependence of ductility on the void volume fraction and the presence of shear lips on the fracture surfaces in our samples suggested a quantitative comparison with our data.

Fig. 4 Scanning electron micrographs (SEM) of a typical fracture surface produced in a tensile test on a specimen containing a disc shaped cavity

The results of the tensile tests presented in Table 1 show that of the three parameters measured, only the ductility (reduction of cross-sectional area) exhibited a sufficiently significant variation over the samples tested to be a meaningful variable. The reason for the small change in ultimate strength with increasing volume fraction of defects is thought to be that the metal near an array of voids may be compared with the necked portion of a tensile bar of ductile material and as Bridgman s has shown large triaxial stresses are induced in the necked region of a deformed bar. While the analogy is not exact, in many respects the region near a bond line containing voids is like the necked tensile bar. The bond-line voids may induce local triaxial stresses of the same type as those found for a cylindrically symmetric tensile bar, though the mathematical form and magnitude will be different. Consequently, while fracture originated at or near the bond-line voids at a strain below bulk-metal failure strain (see for example, McClintock9), the apparent failure stress is not markedly reduced even for significant volume fractions on bond-line voids. One example of some of the configurations is shown in Fig. 4 which is a photograph of the fracture face of a tensile specimen containing a disc-shaped cavity. The detailed view of the fracture surface displays a shear lip around the defect which was typical for all the fracture surfaces examined. No exact theory exists treating the influence of voids or inclusions on the diactility, but several models 1°-13 have recently been proposed. Nagumo 1° has studied ductile fracture of perforated mild steel sheet and has found that the growth of holes was found to proceed rapidly by triaxial stresses associated with the necking of the specimen in accordance with the prediction of hole growth theory. Fracture of the specimen, however, did not result

20

Such a comparison is shown in Fig. 5 which presents tensile data (opencircles) on a graph of ductility as a function of volume fraction of defects. The solid line was obtained by computing the ductility from Nagumo's Equation (9-14) for void volume fractions in the range 0 to 0.1 and fitting the result to the data by the adjuctment of one empirical parameter in Nagumo's model. This parameter relates the shear strain on the crack surface 7 to the mean nominal fracture strain ef of the specimen. Nagumo estimated 3' = 5ef for his systems and we obtained our best fit to the data for 7 = 12ef. This implies that roughly twice as much deformation took place in the very small region just below the crack surface in the Ti-6%A1-4%V than Nagumo estimated for his 0.1% carbon steel sample. Since Nagumo states that his estimate is very rough, and since the Ti-6%A1 -4%V exhibits approximately half the mean nominal fracture strain than his steel, this conclusion does not appear entirely unreasonable. The data shown in Fig. 5 as open circles have been obtained for arrays of voids which were also found to fit the general pattern, but whose interpretation in terms of the back-scattered ultrasonic power is not yet understood in detail. The agreement between the predictions of Nagumo's model and the present data was found not sufficiently good - particularly for the single voids (small volume fraction) to suggest a satisfactory correlation between the ductility and the reflection coefficient. It was therefore decided to find an empirical expression relating the ductility and the radius of the void, while still keeping (for generality) the parameter relating the shear strain on the crack surface to the mean nominal fracture strain of the specimen.

Correlation between ultrasonic and mechanical properties A transcendental equation was found relating the ductility e and the void radius a as follows: exp(½ek)

-

t(l/a) 3 e x p [ - e ( ½

+ k)] =

1 (4)

where k -

7 6f

as discussed above, I is a normalization factor chosen to be the radius of the tensile specimen, and t is a numerical constant.

NDT I N T E R N A T I O N A L . FEBRUARY 1978

nomogram, for example, or by a simpler expression approximating the computer calculations. We have found that for a substantial range of values, Equation (5) is well-approximated by the straight line.

0.5

0 Data for cavities Naqumo theory

0.4 O ~0.

e

o0.

-

e o = S(p

-

S=

"O,--.. O. 0

1 0.05

I 0.10

I

I 0.15

0.20

' Volume fraction' of defects, f

Fig. 5 Comparison o f ductility data f o r cavities in Ti-alloy with theoretical model by Nagumo with k = 12; the data shown here also include some obtained f o r arrays o f cavities (particularly at high values o f f) which are not included in the general discussion

Combining Equations (3) and (4) gives the desired relationship between ductility e and the reflection coefficient p as follows: e x p ( ½ e k ) - t l a c aa [exp(-3p/2B)l [exp(-e(½+k)l = 1 (5) Since Equation (5) is still a transcendental equation, values for e are most easily calculated on the computer and this was done for a wide range of values of p as shown as solid line in Fig. 5. Also shown are data points (open circles) obtained on samples which were first characterized ultrasonically and then subject to tensile tests as described above. A good fit to the data was obtained with a choice o f c = 4 . 3 9 X lO-2(13.8dB),k = 12 in the calculations and t = 1.240.

Discussions and conclusions

t

With the derivation of Equation (5) a relationship has been achieved between the ductility e and the reflection coeffient p with e as the dependent variable. Three fitting parameters have been used. The parameter k describes the ratio of the shear strain on the crack surface 1' to the mean nominal fracture strain ef of the specimen; this parameter is a tensile property of the material and has been estimated by Nagumo to be k = 5 for the case of 0.1% carbon steel and by us to be k = 12 for Ti-6%A1-4%V. The parameter c takes into account losses which the ultrasonic signal suffers in its travel and which are not taken into account by the ultrasonic normalization technique. One error made by the normalization technique is the use of a planar metal/air interface which although being a convenient reference scatterer clearly does not produce the same degrees of beam spreading loss as would a small void. The adjustment of c produces an essentially horizontal shift in the solid line of Fig. 6 (or a shift in Po of Equation (6) discussed below). For simple shapes like spheres, c could be predicted from scattering theory and in fact the shift of 10 dB estimated earlier by comparing theory and experiment of Figs 2 and 3 respectively is not too far from the value (13.8 dB) actually found to give the best fit. The parameter t has its origin in the emperical expression Equation (4) and appears to be close to 1. In practice, Equation (5) is most likely too cumbersome to use since it requires the use of a computer for routine calculations. This problem could be remedied by the use of a

NDT INTERNATIONAL. FEBRUARY 1978

-

Po)

-

- (3/B)[1 - e x p ( - ~ e o k ) l (k/2) + (½ + k)[1 - exp(-keo/2)l

(6)

where S is the slope of the theoretical line at an arbitrary point (eo,#o) selected in the linear regime. The parameter k is the only value which inflences the slope, so that once k is known, values for eo and #o for only one data point are necessary, in principle, to determine the straight line relationship. Then measurements of a give direct information on the ductility of the sample in question. The work to the present, reveals valuable trends in that ultrasonic studies of the amplitude dependence of the scattered sound energy at high frequencies and tensile tests on the same samples appear to correlate in a straightforward way. The relationship between the ultrasonic and mechanical measurements are sufficiently simple to allow a qualitative prediction of the ductility. Thus, these trends suggest the feasibility of predicting the strength of structural members of a technological important unit by ultrasonic defect characterization. The work reported here deals with the simplest and perhaps most important defect, the single large void, whose presence and effect on failure prediction must be the immediate goal of non-destructive evaluation. In addition, however, the single particle or arrays of particles also play important roles and our preliminary investigations show that similar correlations between uniaxial ductility and the scattered sound field are feasible. Consequently the present work assumes much broader implications having significance for example in metallurgical design criteria. In this connection, there has recently been considerable attention placed on the effects of secondary-phase particles on ductile fracture.

\

0.35

~ No defect • Hemisphere (round end towards O Sphere transducer

I[ 0.30

k .

A

~.~ 0.25

a Disc • Cylinder 0 Hemisphere ( with different heat treatment ) Linear approximation

~

0.20

°0.15_

0.10-

0

-50

I

-45

I

"40

l

I

-35

-30

I

-25

I

-20

~I I

-15

l~"~ ~l

-I0

-5

0

Reflection coefficient,p (dB) Fig. 6 Plot o f ductility as a function of the reflection coefficient f r o m data such as shown in Fig. 3 compared to calculations from Equations (5) and (6) for the best fit to the data

21

For example, Argon and co-workers t4'~6 have recently reported analytical and experimental results showing that particles can be detrimental to ductility; they also discuss cavity formation from inclusions in ductile fracture. The importance of ductility has recently been summarized by Tien. ~7 In studies on alloy design with oxide dispersoids and precipitates he concludes that particles are usually not beneficial alloy design elements, if enhanced uniaxial ductility, plane strain ductility, stress rupture ductility and toughness are called for. Such properties as fatigue crack propagation resistance appear to require, for example, both high strength and high ductility, a situation which can come to pass only when the perennial conflict between strength and ductility is resolved in particle strengthened systems in particular and in any other material system in general. That the problem continues to be important in structural applications is seen in the turbine industry for example, which often pays a high penalty in the form of expensive hafnium additions and directional solidification in order to achieve the required minimum ductility in the strong turbine super alloys.

Acknowledgement The author is indebted to the LAAD Division of Rockwell International for partial support of this programme, to J. Gysbers for his valuable help in the computations and to J.M. Richardson for illuminating discussions.

22

References

8

9 10 11 12 13 14 15 16 17

Regalbuto, Y.A. Mater Evaluation 30 (1972) p 66 Vary, A. 'Correlations among ultrasonic propagation factors and fracture toughness properties of metallic'paaterials', NASA TechnicaIMemo TMX-71889 (presented at ASNT 1976 Spring Conference March 8, 1976, Los Angeles, Ca) Tittmann, B.R. Proc IEEE Ultrasonics Symposium (IEEE Cat No 75 CHO 994--4SU, 1975) pp 111-115 Tittmann, B.R. Nadlez, H. and Paton, N.E. Metall Trans 7A (1976) pp 320-324 Hamilton, C.H. 'Ti-Science and Technology', volume 1 (Plenum Press, New York, 1973) p 625 Truell, R., Elbaum, C. and Chick, B. 'Ultrasonic methods in solid state physics' (Academic Press, New York, 1969) pp 161-175 Tittmann, B.R. and Thompl~on, D.O. Mater Evaluation Research Supplement No 35 (1977) pp 74-75 Btidgman, P.W. Trans A S M 32 (1944) p 553 McClintoeh, F.A. J A p p l M e c h 35 (1968) pp 363-371 Nagumo, M. Acta Metall 21 (1972) p 1662 Edelson, B.I. and Baldwin, M.M, Jr, TransASM (1962) p 230 Rice, J.R. and Traeey, D.M. JMech Phys Solids 17 (1969) p 201 also Ttaeey, D.M. Engng Fracture Mech 3 (197l) p 301 and 1 (1971) p 625 Gurland, J. and Plateau, Y. Trans A S M 56 (1963) p 442 Argon, A.S. TransASM98 (1976) pp 60-68 Argon, A.S. and Ira, J.Metall Trans 6A (1975) pp 839-851 Argon, A.S., Ira, J. and Safoglu, R.MetaU Trans 6A (1975) Tien, J.K. Proc o f Batelle Colloqium on Fundamental Aspects o f Structural Alloy Design (held September,, 1975 at Seattle and Harrison Hot Springs, BC, Canada in press

NDT INTERNATIONAL. FEBRUARY 1978