ACI MATERIALS JOURNAL TECHNICAL PAPER Title no. 84-M20

Ultrasonic Pulse Attenuation as a Measure of Damage Growth during Cyclic Loading of Concrete

by Wimal Suaris and Vi raj Fernando

The paper presents experimental results of damage growth during cyclic loading of concrete. The damage growth is inferred from the reduction in amplitude of ultrasonic waveforms transmitted through the specimen during the test. The amplitude of the waveform, by vir-tue of its higher sensitivity to the extent of cracking, is found to be a better indicator of crack growth than the more often used pulse-ve-locity technique. A continuous recording of pulse transmission throughout the tests is obtained by using a pulse generator, a digital oscilloscope, and a microcomputer. A specially constructed frame is used to hold the transducers in contact with the specimen during the test. The damage accumulation during cyclic loading as inferred from the attenuation results is presented for several mix proportions.

Keywords: compression; concretes; cracking (fracturing); cyclic loads; damage; nondestructive tests; ultrasonic tests.

The ultimate behavior of concrete structures under earthquake and other repeated loadings can be pre-dicted by using constitutive relationships obtained for cyclic loading. Most of the experimental results that are available have been obtained by fixing one or more of the test parameters such as the strain-stress level and the loading frequency. 12 Therefore, these results cannot be used directly to predict the ultimate behavior under variable amplitude loading that a structure would sus-tain normally. In fatigue loading situations, attempts have been made to predict failure by using Miner's hy-pothesis. It assumes that the fatigue life consumed dur-ing cycling at a certain stress level is equal to the ratio of the number of cycles to the number of cycles re-quired for failure at that particular stress level. In other words, it means that the deterioration of concrete is proportional to the number of cycles, if cycling is con-ducted at a constant amplitude. Use of Miner's hy-pothesis to predict the fatigue life has often lead to un-conservative results due to this assumption. 1

As continuous microcrack growth occurs during cy-cling, measurement of crack growth presents a very in-tuitively appealing method of monitoring the deterio-ration that occurs during cycling. The results would also indicate why a linear degradation assumption may be unconservative.

ACI Materials Journal I May-June 1987

A recorded history parameter such as cracking may also be used to predict the remaining life of an in situ structure, by using a combination of in situ measure-ments and calibration curves obtained for laboratory specimens. The prediction of remaining life of in situ structures has been receiving much attention as the na-tion's infrastructure deteriorates due to overuse, envi-ronmental effects, and aging.

The measurement of cracking has to be conducted by the use of nondestructive techniques in order to obtain a continuous measurement of crack growth during cyclic loading. In the past, various methods such as acoustic emission, infrared thermography, pulse echo, and a host of magnetic and electrical methods have been used to determine the extent of cracking. 3 Out of all these methods, the ultrasonic methods offer the dis-tinct advantages of the ability to determine the interior properties of the specimen, the detection of deteriora-tion even after the occurrence, and the possibility of taking measurements without causing any damage and at relatively low cost. In the present study ultrasonic attenuation was used instead of ultrasonic pulse veloc-ity to monitor crack growth because of the higher res-olution offered by the attenuation method.

The paper presents results from a number of uniaxial compression and cyclic loading tests conducted on sev-eral concrete mixes. Crack growth data inferred from ultrasonic measurements are given for all specimens.

RESEARCH SIGNIFICANCE This paper demonstrates the feasibility of using ul-

trasonic pulse attenuation as a tool for assessing the extent of crack growth during cyclic loading of con-crete. The results indicate that crack growth is not lin-early proportional to the number of load cycles, which

Received May 27, 1986, and reviewed under Institute publication policies. Copyright 1987, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright propri etors. Pertinent discussion will be published in the March-April 1988 ACI !vfateria/s Journal if received by Dec. I, 1987.

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ACI member Wimal Suaris is an assistant professor of civil engineering at the University of Miami. He is an active researcher in the area of cracking and fracture of concrete. He is a member of ACI Committees 224, Cracking, and 446, Fracture Mechanics. He is also a member of the ASCE-EMD Committee on Experimental Analysis and Instrumentation.

Viraj Fernando is a graduate research assistant at the University of Miami, where he is pursuing his PhD degree. He is currently on leave from the Univer-sity of Sri Lanka, where he is an assistant lecturer.

may help explain why methods such as Miner's hy-pothesis would lead to unconservative fatigue life pre-dictions under variable amplitude loading. The damage growth curves obtained from laboratory tests may be used to predict the extent of deterioration in existing structures that have already undergone some damage due to cycling. Although the present study was only conducted to monitor crack growth during cyclic load-ing, the technique proposed may conceivably be used to determine the extent of damage due to creep, shrink-age, and other environmental effects.

NONDESTRUCTIVE CRACK MONITORING TECHNIQUES USED FOR CONCRETE

Nondestructive techniques that have been used to monitor crack growth in concrete include ultrasonic waves, acoustic emission, infrared thermography, and a host of other magnetic and electrical methods. 3 In the acoustic emission technique, the low-frequency sound emitted during the formation and growth of cracks is monitored. The location and the orientation of the cracks may be monitored by using a suitable arrange-ment of transducers. 4 The major drawback of this tech-nique is that cracks can be monitored during their for-mation process only. Most of the other proposed non-destructive techniques yield information only on the surface properties and interior properties cannot be ob-tained.

Ultrasonic waves Ultrasonic waves are mechanical waves with fre-

quencies usually in excess of 16 kHz. These waves are generated by exciting a piezo-electric crystal with a high voltage pulse. The wave is then transmitted through the test material, which is in contact with the transducer containing the crystal. When this wave impinges upon a receiving transducer it produces an output voltage. Depending on the cut of the crystal and the coupling used, the waves transmitted can set up compressional-dilational stresses or shear stresses as they pass through the material. Accordingly, these two types of waves are referred to as pressure waves and shear waves. In the concrete industry, compression waves have been used exclusively for ultrasonic testing, apparently because of the difficulty of coupling shear transducers and be-cause of the higher attenuation characteristics of shear waves.

Ultrasonic techniques used for concrete Ultrasonic pulse echo methods have been employed

to locate finite size flaws in concrete. 56 Waves that are reflected at the crack interfaces affect the received

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waveform. The locations and dimensions of the flaws may be determined by analyzing the waveforms re-ceived for several positions of the transducers. This technique cannot, however, be used to detect a large number of small distributed cracks. Distributed cracks of this nature exist in concrete prior to loading and also during load application until they finally coalesce at the point of failure.

Ultrasonic transmission methods have, therefore, generally been used to assess the quality of concrete. In this method the receiving transducer is placed on a specimen surface different from that of the transmit-ting transducer. The pulse transit time between the two transducers is governed by the fastest wave received which is always the direct wave for a homogeneou~ material. The transit time is measured by electronic timing circuits and usually appears on a digital display. Many attempts have been made to correlate the pulse velocity calculated from the transit time to the com-pressive strength of concrete. It has been found, how-ever, that even if such a relationship can be estab-lished, it would be strongly dependent on factors such as materials used, curing conditions, moisture content, etc. 7-8 Furthermore, it has also been observed that a re-lationship established during the developmental stages of the concrete cannot be used to predict the strength of concrete that has already undergone some deterio-ration. As a result, little confidence has emerged among engineers on the use of ultrasonic inspection techniques for predicting the compressive strength of concrete. Ul-trasonic attenuation techniques can, however, be used with more success to assess the deterioration in a con-crete member.

Shah and Chandra9 and Raju'0 have conducted un-iaxial compression tests while monitoring the ultrasonic pulse velocity in a direction perpendicular to the axis of loading. The longitudinal cracks that developed under uniaxial compression were expected to decrease the lat-eral pulse velocity. The researchers found, however, that the pulse velocity did not change appreciably until about 90 percent of the peak compressive stress was reached. As microscopic studies indicate that crack growth begins at a much smaller stress level, a more sensitive indicator of crack growth is required.

The amplitude of the ultrasonic waveform decreases gradually as it passes through the specimen. This atten-uation can be used to monitor the crack growth in a specimen, if the transmission length is kept constant and the transducers are acoustically well coupled with the specimen. The attenuation would then depend on the frequency of the transducer and the size and num-ber of flaws encountered between the transmitter and the receiver. In the present study this ultrasonic atten-uation was found to be a much more sensitive indicator of crack growth than pulse velocity.

EXPERIMENTAL PROGRAM Specimen preparation

The laboratory tests were conducted by using 3 in. diameter and 6 in. high specimens. The materials used

ACI Materials Journal I May-June 1987

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Table 1 .- Mix proportions and basic mechanical properties

Axial Compressive

Initial pulse tangent velocity,

strength f' , modulus ""'' Poisson ft/sec, Designation Mix proportion psi psi ratio 700kHz

A I :2:3 (0.5) 5650 4.7 X 10" 0.19 17,900 B 1:2:3 (0.4) 6100 5.2 X 10'' 0.18 18,050 c 1.2.4 (0.45) 5925 5.0 X 10' 0.18 18,100 I ps1 6.9 kpa, I ft/sec - 0.305 m/sec

were Type I portland cement conforming to ASTM ~pecification C 150, river sand passing through U.S. su~ve No. 10, and \12 in. (12. 7 mm) river gravel. The mix proportions investigated in the study are given in Table 1. One batch was made for each mix proportion. The free water-cement ratio given in the table was cal-culated after allowing for the absorption of water by th~ aggregates. The mixing was conducted in a rotary m1x~r and the compaction was carried out using a vi-brating table. The concrete was cast in waxed card-board molds and kept under ambient conditions for 24 hours, covered with a polythene film. The molds were stripped and the specimens were cured in a humidity room for 28 days before testing them.

Strain gages were installed on some of the specimens at diametrically opposite points in both the axial and lateral directions. Standard wire strain gages with a plastic backing and impregnated with a polyester resin were used. The gages had a length of 1.18 in. (30 mm) and a gage factor of 2.09. The specimen surface where the strain gages were to be installed was prepared by abrading it with sandpaper and then cleaning it with acetone. A polyester adhesive was used to bond the strain gages to the specimen surface. A clamping pres-sure was also applied on rubber pads placed over the strain gages until the adhesive set.

Ultrasonic equipment The ultrasonic pulse transmission during the tests

were monitored in both the axial and lateral directions using P-wave transducers. Specially constructed load-ing platens with 700 kHz crystals embedded in them were used to transmit and receive pulses in the axial di-rection. Lateral pulse transmission was monitored in two diametrically opposite directions using two pairs of transducers. One transducer pair had a crystal fre-quency of 500 kHz while the other had a frequency of 150kHz. Using two separate frequencies was expected to provide better resolution than using a single fre-quency, as the level of attenuation depends on the crys-tal frequency. The transducers were interfaced to curved metal pads having a silicone rubber facing that matched the curvature of the specimen. These lateral transducers were mounted on a circular frame and spring loaded to accommodate the lateral expansion of the specimen during loading. An ultrasonic couplant was applied between the specimen and all transducers to facilitate better pulse transmission. Fig. 1 is a pho-tograph of the transducer ring.

ACI Materials Journal I May-June 1987

Fig. 1-U/trasonic transducer ring

_Th~ excitation for the transducers was provided by a seismic analyzer. A digital display in the seismic ana-lyzer indicated the pulse transit time between the trans-mitting and receiving transducers. The received wave-form, after being amplified by the seismic analyzer, was passed on to a digital storage oscilloscope. The digital oscilloscope was connected to a microcomputer via a RS-232c interface, which enabled the storage and the subsequent retrieval of the waveforms.

Testing procedure Compression tests under a monotonically increasing

compressive stress were conducted for two specimens in each concrete mix. The tests were performed using a 55 kip hydraulic actuator and loading frame. All tests were conducted at a constant strain rate of about 10 6/sec in the stroke control mode, using a ramp wave signal gen-erator.

Cyclic loading tests were performed for a minimum of three concrete specimens in each concrete mix. The tests were conducted by using a function generator to cycle the load sinusoidally between preset limits. In most of the tests the maximum stress was adjusted to 1.5 times the mean stress and the minimum stress was adjusted to 0.5 times the mean stress. Some tests were also conducted with a minimum stress close to zero. The maximum stress values investigated varied from 85 to 97 percent of the monotonic compressive strength. These high peak stresses were chosen to insure the fail-tire of the specimen within about 500 cycles. All tests were conducted at a frequency of 0.01 Hz.

The strains during the tests were monitored by using a data acquistion system. The two axial and the two lateral strain gages were connected to four separate channels of the system. Each strain gage was arranged as a quarter bridge circuit and a three lead wire system was used to compensate for any temperature effects. The strain readings were taken at several stress levels during the first cycle and then in the vicinity of the maximum, mean, and minimum stresses during subse-quent cycles. The output from the load-cell transducer was also connected to one channel of the data acquisi-tion system.

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Data Acquisition Systan

strain Gage seamer

Fran Spr:ing Load Cell

Platen

Fig. 2-Schematic diagram of test setup

Voltage pulses to the transmitting transducer were generated by the seismic analyzer at a frequency of 500 Hz. This frequency insured that the waveform from the receiving crystal attenuated completely before the next pulse was transmitted. The output pulse monitor was connected to one channel of the oscilloscope and was also used as the triggering source. The amplified out-put from the receiving transducer was connected to the other channel of the oscilloscope. Thus, when the os-cilloscope was triggered by the input pulse, the wave-form from the receiving transducer appeared on the screen beginning at a time corresponding to the transit time between the transducers. The sampling rate was selected as 50 nanoseconds, which resulted in a total data window of 100 microseconds. Thus, the display contained a minimum of 10 cycles of the received waveform for the frequencies used.

The computer was programmed to trigger the oscil-loscope and display the received waveform when any keyboard command was entered. The waveform was then stored in the computer. The capturing and trans-ferring of the waveform was accomplished in less than 5 seconds. As two lateral transducer pairs were used, the waveform to be recorded was selected by using a switch. The waveforms were recorded at the same points at which strain readings were taken.

A schematic diagram of the test setup is given in Fig. 2. Connections to only one pair of lateral transducers are shown in the figure, although two transducer pairs were used during the actual test.

DISCUSSION OF TEST RESULTS Uniaxial compression

The results from the uniaxial compression tests with a monotonically increasing load are summarized in Ta-bles I and 2. The reduction in secant modulus has usu-ally been associated with the microcracking of concrete by many past researchers.

The pulse velocity values given in the tables were cal-culated by dividing the distance between the trans-ducers by the transit time. The axial pulse velocity was found to increase slightly at the beginning of load ap-plication, apparently because of ~etter transducer c~ntact and the closure of cracks/vmds due to the apphed compressive stress. Only a marginal decrease in the ax-ial velocity at the peak stress was observed. This can be expected, as crack growth would predominantly be parallel to the axis of the specimen.

The lateral pulse velocity is about 25 percent less than the axial pulse velocity. This difference may be attrib-uted to the better compaction in the axial direction as the specimens were cast vertically. The frequency of the transducer used for lateral pulse transmission was also lower than the frequency of the transducer used for ax-ial pulse transmission. Others have also observed trans-mission velocities to reduce with a decrease in trans-ducer frequency. 9 The lateral pulse velocity was found to remain essentially constant up to about 90 percent of the peak stress. It decreased only by about 12 percent when failure was imminent. The pulse velocity is there-fore not a very sensitive indicator of crack growth, as observed by others. 910 Moreover, it was also observed that the point of decrease in lateral velocity coincided with the point where the lateral strains began to in-crease sharply, implying that the decrease in pulse ve-locity may be due to the opening of the cracks rather than actual crack growth itself.

Ultrasonic attenuation The microcracks that develop during a uniaxial

compression test affect other features of the ultrasonic waveform in addition to the transit time. Fig. 3 shows typcial waveforms received from the lateral transducers at the beginning of the test and after applying a certain amount of load. Comparing the two waveforms, it can

Table 2 - Summary of ultrasonic measurements obtained from uniaxial compression tests u/um,, = 0.0 u/umu = 0.90 u/u""'' = 0.99 Lateral pulse Pulse velocity Reduction in Secant Pulse velocity Reduction in Secant velocity ft/sec reduction, percent amplitude, percent modulus reduction, percent amplitude, percent modulus

Designation 150kHz 500kHz 150kHz 500kHz 150kHz 500kHz E,.,, psi 150kHz 500kHz 150kHz 500kHz E_,., psi A 13,500 13,100 2.9 2.0 9 29 4.0 X 10" 5.9 9.8 55 48 3.5 X 10" B 13,725 13,570 1.9 1.5 12 19 4.7 X 10" 6.0 7.8 41 55 4.0 X 10' c 13,725 13,300 1.5 2.3 14.5 15 4.3 X 10" 4.0 8.4 58 67 3.8 X 10'

I psi = 6.9 kpa

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Fig. 3-U/trasonic waveforms received from an un-stressed and stressed (90 percent of peak stress) speci-men

be noticed that the amplitude of the waveform has undergone a significant reduction after load applica-tion.

-

80 A - 1:2:3; 0.5 (/} B - 1:2~3; 0.4 Ul c - 1:2:3; 0.45 ~ (/) ~ 60 & 4-< 0

""

~ 40 Ul Ul

~ (/) 20

0 0.2 0.4 0.6 0.8 1.0

a:MBINED DAMAGE COEFFICIENT

Fig. 5- Variation of combined damage coefficient for different concrete mixes during uniaxial compression

(a)

(c)

be noticed for all specimens. The damage coefficient calculated using both frequencies and Eq. (2) is also shown plotted with the stress level in Fig. 5. The curves given in Fig. 5 represent the average of two specimens tested. Note that the results for all the mix proportions fall close together. The attenuation rate thus appears to 1 be independent of the mix proportions for at least those tested here.

Cyclic compression tests Ultrasonic waveforms received at four stages during

a cyclic loading test are shown in Fig. 6. The first through fourth waveforms are arranged in the ascend-ing order of number of load cycles. It can be noticed by comparing the final waveform with the initial wave-form that the initial high frequency waves have been almost completely attenuated and the lower frequency waves have become predominant. If an oscilloscope was not used, this could lead to erroneous pulse transit time readings. This is because the digital displays in pulse velocity measuring instruments have a timing cir-cuit that is triggered when the waveforms exceed a cer-tain magnitude. The attenuation of the waveform be-

(b)

(d) Fig. 6-U/trasonic waveforms received during four stages of cyclic loading using 500kHz transducers (number of cycles to failure = 338): (a) under no stress; (b) after completing one cycle; (c) after completing 168 cycles; (d) after completing 329 cycles

190 ACI Materials Journal I May-June 1987

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80

No. of Cycles N

1 or-----------~1o~----------~w~2----------~103

(1.8-5.5 ksi)

1oo.__ _____________________ __,

Fig. 7-Percentage reduction in amplitude of the 150 kHz frequency during cyclic loading (concrete mix pro-portion = 1:2:3;0.5; 1 ksi = 6.9 MPa)

low this level would cause the timing circuit to be trig-. gered at a subsequent peak, resulting in a higher transit time being displayed.

The attenuation versus number of load cycles for the two transducers is shown in Fig. 7 and 8 for specimens tested from one batch. The minimum and the maxi-mum stresses during the cycling are indicated in the figure. It can be noticed that a higher relative attenua-tion during the initial cycle generally results in failure within a relatively smaller number of cycles. However, the results from the 500 kHz transducer show that Specimen A6 failed after a relatively shorter number of cycles despite the lower initial attenuation during the initial cycle. The reason for this becomes clear when the very high attenuation during the initial cycle of the 150 kHz transducer is noted. The specimen thus appears to have developed more cracks in planes perpendicular to

1.0

AS (1. 7-5.1 ksi) +' A3 (1.7-5.2 ksi) m M u

.-<

""' ""'

f 0.5

A6 (1.8-5.5 ksi) A4 (1.8-5.3 ksi)

"@ 0.2 -~ ~

Q 20 40 60 80 100

(N/Nf) X 100

Fig. 9- Variation of combined damage coefficient with cycle ratio (concrete mix proportion = 1:2:3;0.5; 1 ksi = 6.9MPa) ACI Materials Journal I May-June 1987

No. of Cycles N

Fig. 8-Percentage reduction in amplitude of the 500 kHz frequency during cyclic loading (concrete mix pro-portion = 1:2:3;0.5; 1 ksi = 6.9 MPa)

the transmission path of the 150 kHz transducer. This clearly illustrates the importance of monitoring crack growth in two directions when axisymmetric conditions are present. The number of cycles up to failure is also found to be inversely related to the maximum stress, except for Specimen A5 The higher level of attenuation for a lower stress during the first cycle suggests, how-ever, that this particular specimen was weaker than the other specimens.

Fig. 9 through 11 show the variation of the com-bined damage coefficient with cycle ratio for all the specimens tested from the three concrete mixes. It ap-pears that the damage accumulation during cyclic load-ing is virtually independent of the maximum stress level when the results are presented as a function of the non-dimensional cycle ratio. The scatter in the results is generally similar to other cyclic loading tests.

1.0

~ B3 (0.4-5.6 ksi) . .., BS (0.4-5.2 ksi) u a. 75 . .., ""'

B4 (0.4-5.8 ksi) ""'

I 0.50 al

-~ 0.25 ~ 20 40 60 80 100

(N/Nf) X 100

Fig. 10- Variation of combined damage coefficient with cycle ratio (concrete mix proportion 1:2:3;0.4; 1 ksi = 6.9 MPa)

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" 0.75 ] f 0.50

0

C3 (2.1-5.8 ksi) --------.,....-::;:~'/" cs (2.0-5.5 ksi) ----:7"

tion during cyclic loading is not linear, particularly during the initial cycles. Therefore, using methods based on linear damage accumulation (such as Miner's hypothesis) may lead to unconservative results when predicting fatigue life under variable amplitude load-ing. The damage accumulation rate was not found to be significantly related to the concrete mix proportions used. The mix with the lower water-cement ratio had a slightly smaller damage growth rate, apparently be-cause of the smaller number of initial cracks. Although the authors have not undertaken it in the present study, the damage accumulation inferred from the ultrasonic readings may be verified by conducting microscopic observations of specimens that have been subjected to different amounts of cycles.

ACKNOWLEDGMENTS The research was partially supported by National Science Founda-

tion Grant No. CEE8404713 to the University of Miami. The authors also acknowledge funds provided for equipment purchases by the University of Miami.

A A,

A

A,

NOTATION amplitude of waveform received from the stressed specimen amplitude of waveform received from the unstressed speci-men

vector with the amplitude of the waveforms (150 kHz, 500 kHz) as components, received from the stressed specimen vector with the amplitude of the waveforms (150 kHz, 500 kHz) as components, received from the unstressed specimen secant modulus initial tangent modulus uniaxial compressive strength compressive stress number of cycles number of cycles to failure

REFERENCES I. Holmen, Jan Ove, "Fatigue of Concrete by Constant and Vari-

able Amplitude Loading," Fatigue of Concrete Structures, SP-75, American Concrete Institute, Detroit, 1982, pp. 71-110.

2. Tepfers, Ralejs, and Kutti, Thomas, "Fatigue Strength of Plain, Ordinary. and Lightweight Concrete," ACI JouRNAL, Proceedings V. 76, No. 5, May 1979, pp. 635-652.

3. Malhotra, V. M., "In Situ/Nondestructive Testing of Con-crete-A Global Review," In Situ/Nondestructive Testing of Con-crete, SP-82, American Concrete Institute, Detroit, 1984, pp. 1-16.

4. Maji, A. K., and Shah, S. P., "A Study of Fracture Process of Concrete Using Acoustic Emission," Proceedings, Spring Confer-ence (New Orleans, June 1986), Society for Experimental Mechanics, Bethel, pp. 1-13.

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80

i ~ 40

0 20

Miner's Hypothesis

--J'"'-=--"""':,.,e:..- 150 kHz Reduction in ~~-- 500 kHz Amplitude

40 60 80

(N/Nf) X 100 100

Fig. 15-Damage accumulation for Specimen B4 based on different methods

5. Carino, Nicholas J.; Sansalone, Mary; and Hsu, Nelson N., "A Point Source-Point Receiver, Pulse-Echo Technique for Flaw Detec-tion in Concrete," ACI JoURNAL, Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 199-208.

6. Krautkramer, J., and Krautkramer, H., Ultrasonic Testing of Materials 3rd Edition, Springer Verlag, New York, 1983, 667 pp.

7. Facaoaru, I., "Non-Destructive Testing of Concrete in Ro-mania," Proceedings, Symposium on Non-Destructive Testing of Concrete and Timber (June 1969), Institution of Civil Engineers, London, 1970, pp. 39-49.

8. Sturrup, V. R.; Vecchio, F. J.; and Caratin, H., "Pulse Veloc-ity as a Measure of Concrete Compressive Strength," In Situ/Non-destructive Testing of Concrete, SP-82, American Concrete Institute, Detroit, 1984, pp. 201-227.

9. Shah, S. P., and Chandra, S., "Mechanical Behavior of Con-crete Examined by Ultrasonic Measurements," Journal of Materials, V. 5, No. 3, Sept. 1970, pp. 550-563.

10. Raju, N. K., "Small Concrete Specimens Under Repeated Compressive Loads by Pulse Velocity Technique," Journal of Mate-rials, V. 5, No. 2, June 1970, pp. 262-272.

II. Rose, J. L.; Nestleroth, J. B.; and Jeong, Y. H., "Component Identification Using Ultrasonic Signature Analysis," Journal of Ma-terials and Evaluation, V. 41, Mar. 1983, pp. 315-318.

12. Suaris, Wimal, and Shah, Surendra P ., "A Rate-Sen~itive Damage Theory for Brittle Solids," Journal of Engineering Mechan-ics, ASCE, V. 110, No.6, June 1984, pp. 985-997.

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