Technical note

Acoustoelastic determination of local surfacestresses in polymethylmethacrylate

Zhi Wei a,*, Xiaojun Zhou b, Yaodong Cheng b

aSchool of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, PR ChinabDepartment of Mechanical Engineering, Zhejiang University, Hangzhou 310027, PR China

Received 13 May 1999; received in revised form 24 November 1999; accepted 10 February 2000

Abstract

Acoustoelastic determination of applied and residual stresses plays an important role as a

nondestructive tool in current engineering. Rayleigh surface wave, as a special form of ultra-sonic waves, makes it possible to probe initial stress at various depths from the surface bychanging its frequencies because of its propagation characteristic. In this paper, the propaga-

tion of Rayleigh surface wave in isotropic elastic material subjected to a homogeneous stressis studied theoretically in view of elastic wave and ®nite deformation. Speci®c surface ultra-sonic transducers are constructed to take the measurement of surface stresses and surfacecracks in components with very low ultrasonic velocity such as polymethylmethacrylate. The

linear dependence of velocity change of Rayleigh ultrasonic wave on the applied static stressis used in determination of local surface stresses in the specimenmade of polymethylmethacrylate.Both the acoustoelastic and gauge experiments are synchronously carried out for a specimen

with a central hole under uniaxial tensile stress. The results and the experimental errors areshown and analyzed. # 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Polymers are widely used to make components in aerospace vehicles, automobilesand ships, etc., such as PMMA for aerospace transparent component. They are verydi�erent from metals in their low elasticity and low strength. They are quite sensitiveto defect and stress concentration and poor at resistance to crack extension. Therefore,they are vulnerable to the hazard caused by both residual and working stress. Hence

Applied Acoustics 61 (2000) 477±485

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* Corresponding author. Tel.: +86-22-2656-4224 (o�ce), +86-22-2603-2144 (home); fax: +86-22-

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E-mail address: weiwhizz@263.net (Z. Wei).

it is vital to be able to evaluate the polymer's working condition and service life. Tosolve the problem relies heavily on nondestructive testing (NDT) of the component.As there is a shortage of e�ective testing methods there is a problem of waste ofmaterial and the hidden danger of accident.It is well known that much of the destruction or failure in polymer components is

caused by surface cracks and/or surface stresses. Ultrasonic surface wave testing isone of the most e�ective methods of detecting surface defects and evaluating surfacestresses. Therefore, the pertinent transducer (oblique probe) plays one of the mostsigni®cant roles in dealing with the above problems. The fact that ultrasound pro-pagates slower in a polymer component than in metal makes it di�cult to obtainsurface acoustic wave in the component by means of the oblique probe because therehas to be a particular acoustic entrant material for the wedge in which longitudinalultrasound propagates much slower than the transverse one in the component. It isunfortunate that present materials including metals and non-metals rarely have theindispensable physical performance and chemical property required as the acousticentrant material, in terms of ultrasound velocity, acoustic attenuation, hardness,elasticity, high temperature resistance, and corrosion resistance, etc. In this paper, inview of the practical use of ultrasonic transducers, the mechanism of ultrasoundvelocity and attenuation in a medium is investigated. A kind of room temperaturevulcanized (RTV) silicone elastomer is developed as acoustic entrant material, andthe corresponding surface wave transducers are designed for ultrasonic NDT ofsurface stress and crack in a transparent aerospace component. By means of thosetransducers a set of experimental equipment is designed for a series of acoustoelasticdeterminations of surface stress in PMMA components.

2. Theory of acoustoelasticity of Rayleigh surface waves

The stress ®eld in the material through which they travel in¯uences the velocity ofultrasonic waves. This dependence could be a very useful tool in determining thestress present in the material by measuring the velocity variation. Several authorshave proposed a variety of ultrasonic techniques to measure the acoustoelastic e�ectin metals using bulk or surface waves [1±6]. In this paper attention is focused onRayleigh surface waves travelling in polymethylmethacrylate (PMMA).Acoustoelasticity refers to the dependence of the wave velocity on stress or

deformation. Developments in acoustoelasticity have been stimulated by interest inthe measurement of third-order elastic constants and residual stresses. According todiscussions about surface acoustoelastic e�ects conducted by Hirao, Allison, Tver-dokhlebov, and Makhort [4,6,7,8], when the Rayleigh wave propagates on the freesurface of a semi-in®nite homogeneous isotropic material initially under uniaxialstatic stress, shown in Fig. 1, the dependence of the ultrasonic velocity � on the staticstress � can be described in the form

vÿ v0v0� K� �1�

478 Z. Wei et al. / Applied Acoustics 61 (2000) 477±485

where �0 represents the Rayleigh wave velocity in the natural state of the materialunder discussion, and K is an acoustoelastic coe�cient governed by material con-stants such as density, elastic modulus, etc.

3. Surface ultrasonic transducer

3.1. Material for wedge

As is known, only if the longitudinal ultrasound propagates slower in wedge thantransverse one does in the component can a surface wave be obtained in it when thewave model conversion based transducer is used. So the transverse ultrasound pro-pagates at about 1100 m/s (when frequency is 1 MHz) in PMMA [9], for instance,the longitudinal wave ought to travel at less than 1000 m/s in the acoustic entrantmaterial for this application.In spite of the fact that acoustic attenuation is helpful in reducing clutter re¯ection

in the transducer, excessive attenuation may consume acoustic energy, which cer-tainly in¯uences accuracy and reliability of testing. Polymer materials generallyabsorb acoustic energy more than metals. So very low attenuation is favorable in anultrasonic transducer. The problem of clutter re¯ection can be overcome throughcareful design of transducer structure.After many trials, a new kind of RTV silicone elastomer was prepared with visc-

osity 9000 to 14 000 CS (under 25�C). In the trials, polydimethyloxosilane is theprincipal raw material. Silicone elastomer with di�erent molecular weights is mixedin di�erent ratios with curing agent (1±3%), cross-linking agent (1±5%) and packingmaterial. Having been uniformly agitated it is successfully vulcanized at room tem-perature for 3±4 h after vacuum degasi®cation and ®nally ripening at about 30�Cfor 40±50 h. In this material longitudinal ultrasound can propagate at a speed of951±988 m/s with attenuation coe�cient lower than 0.5 dB/mm.

3.2. Transducer structure

For ultrasonic NDT of surface stress and ®ne cracks in polymer material such asaerospace transparent components, the size and frequency of the piezoelectric crys-tal plate (wafer) must be so selected that the transducer is able to scan the range

Fig. 1. Rayleigh wave propagation on surface of a uniaxial stressed solid.

Z. Wei et al. / Applied Acoustics 61 (2000) 477±485 479

widely, to ®nd the ®ne defect e�ectively and to transmit the echo strongly. Thereforethe size of the wafer should be as large as possible and rectangularly shaped. In thispaper three types of PZT wafers are designed with size of 8�10 mm2 and frequency5 MHz.The shape of the wedge has to meet the requirement of incident angle. For exam-

ple, if the velocity of longitudinal ultrasound is 1000 m/s in the wedge, there can be asurface acoustic wave in the PMMA component only when the incident angle is inexcess of 64�. The larger the incident angle is, the further the ultrasound propagatesin the wedge. This certainly brings about problems concerning both acoustic energyloss and size of transducer. Hence, an incident angle of 65� is chosen while theultrasonic transducer is fabricated.Owing to the fact that acoustic pressure changes sharply in the near ®eld and there

can hardly be a stable re¯ecting echo, the tested region ought to be beyond the near®eld. Because of the fact that it is di�cult to set acoustic traps, the shape of thewedge is designed as shown in Fig. 2 mainly in order to reduce clutter re¯ection.

3.3. Fabrication of the transducer

Because the RTV silicone elastomer is limited by sti�ness and hardness, it is di�-cult to bind the wafer with the wedge by an adhesive. For ®rm contact of the waferwith the wedge, the RTV silicone in solution is poured into the prefabricated con-tainer when lead wire, socket as well as outer shell are solidi®ed together after thewafer has been ®rmly bound with backing material for sound absorption.

4. Calibration test

In order to obtain the stresses using Eq. (1), v0 and the constant K can be deter-mined experimentally by a calibration test.To determine �0, the travelling time over di�erent distances have been measured

for 5 MHz transducers mounted on a rectangular PMMA specimen in a stress-freestate. To calculate �0 from measured intervals, the following equation is used.

v0 � di�1 ÿ diti�1 ÿ ti

�2�

Fig. 2. Shape of the wedge.

480 Z. Wei et al. / Applied Acoustics 61 (2000) 477±485

where di and ti are the physical distance between two transducers (one as transmitterand another receiver) and the time interval between two peaks of wave signals,respectively. The average value of �0 is 1076 m/s for the condition that di varies from10 to 22 mm with a dispersion of 2 mm.A 3 mm PMMA YB-3 plate has been cut into a dumbbell shape, as shown in

Fig. 3. Transducers are freely laid 12 mm apart on the surface of the specimen. Thespecimen is loaded in a manual-loading device specially designed for the experiment.The velocities of the surface wave in the load direction are measured at each loadingstep.The measurement device is connected to a broadband ultrasonic pulser-receiver

which excites the transmitter and receives and ampli®es the signals coming from thereceiving probe. The pulser has been set to excite the transmitter with a repetitionrate of 200 kHz. The received pulses are ampli®ed 30 dB and then displayed on adigital oscilloscope with 100 MHz sampling frequency. The digital oscilloscope isconnected to a personal computer where the waveforms are stored. After theacquisition of the signal coming from the receiver, the time interval can be acquiredby the computer and mathematically operated on. The measurement system isshown in Fig. 4.When the specimen is statically loaded, if Rayleigh surface wave with velocity �

travels a distance d in time t, we have

vÿ v0 � ÿ�tÿ t0�v0�tÿ t0� � d=�0

�3�

Fig. 3. Specimen for a calibration test in uniaxial loading.

Fig. 4. Block diagram of experimental set-up.

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where t0 is the time the Rayleigh surface wave spends travelling the same distancewith �0 in the specimen. Thus, the velocities of Rayleigh surface wave propagation ineach loading step can be calculated by Eq. (3) for every detected time interval.The measured dependence of relative velocity changes on stresses for the specimen

in uniaxial tensile loading test are shown in Fig. 5, where the stresses are calculatedaccording to the load and the cross area in the middle of the specimen. This impliesthat under the experimental condition the wave velocity varies almost linearly withtensile stress, so we can describe this acoustoelasticity approximately by dint of alinear relation. Using least-square-®tting of these data, the acousoelastic coe�cientin Eq. (1) for PMMA YB-3 are obtained as K � ÿ0:00288 MPaÿ1 and Eq. (1) renders

vÿ v0v0� 0:00259ÿ 0:00288� �4�

5. Measurement of local surface stresses

A PMMA YB-3 specimen with a central circular hole has been machined, asshown in Fig. 6. Gauges are mounted on the surface along AB and two surface wavetransducers are freely laid across AB on the opposite surface of the specimen asshown in Fig. 7.A tensile load is applied to the specimen at constant stress level in the experiment.

The experimental set-up is almost the same as shown in Fig. 4 except for the addition

Fig. 5. Stress dependence of velocity of Rayleigh wave parallel to the stress direction.

482 Z. Wei et al. / Applied Acoustics 61 (2000) 477±485

of a strain-gauge board. With load 900 N, Table 1 lists the detected relative velocitychanges and strains, both parallel to the loading direction for di�erent points. Fig. 8shows the stress distribution along AB in the specimen under this load. In the ®gure,stresses illustrated by hollow circular symbols are calculated according to Eq. (4) fordetected data. The solid ones show results approximately evaluated by strains listedin the last column of Table 1

�x � ��3l� 2��l� � "x �5�

where the Lame elastic constants have been measured to be be l � 5:718 GPa, � �1:422 GPa [10]. The solid curve represents a geometric regressing result of the datadetermined by acoustoelastic measurement, which has the following form

�x � 63:9yÿ0:76 �MPa� �6�

If the specimen were in®nite, according to theory of elasticity, stress distributionalong AB can be given by

Fig. 6. Specimen for test of local surface stress concentration.

Fig. 7. Schematic diagram for arrangement of strain gauges and transducers.

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�x � �02�2� �

2

y2� 3�4

y4� �7�

where � is the radius of the hole and �0 the average stress evaluated according to theload and middle cross-area of the specimen. In the experimental condition it has tobe about 6.67 MPa. Hence, we obtain the theoretical curve in Fig. 8 described by adiamond shape.

Fig. 8. Stress distribution in the direction of load.

Table 1

Coodinates and data of relative velocity changes and strains

Coordinates (mm) Strain "x (10ÿ3) Relative velocity change �ÿ �0� �=�0 (10ÿ2)x y

0.0 5.5 3.79 ÿ4.670.0 8.0 3.06 ÿ3.830.0 10.5 2.44 ÿ2.560.0 13.0 2.23 ÿ2.360.0 15.5 2.03 ÿ1.990.0 18.0 1.53 ÿ1.810.0 20.5 1.35 ÿ1.500.0 23.0 1.33 ÿ1.410.0 25.5 1.30 ÿ1.35

484 Z. Wei et al. / Applied Acoustics 61 (2000) 477±485

6. Discussion and conclusions

It is seen in Fig. 8 that the local stresses near the brim of the circle hole, measuredby means of both acoustoelasticity and strain gauge, are about 3 times that of thosedistant from the hole, i.e. near the side of the sample. This ratio is in agreement withthe theoretical prediction of Eq. (7). The result gives a clear indication that the sur-face stress concentration falls rapidly with distance from the hole.However, it is clear that the value of measurement data is a little di�erent from

that of theoretical result, we believe this is mainly attributed to two facts, one is thattwo detected results, by ultrasound and gauge respectively, have all neglected thein¯uence of stress perpendicular to the loading direction, and another the di�cultyof accurately locating both gauge and transducer especially at the brim of the hole.Despite this, the experimental results do not render excellent accuracy because the

transducers are moved manually perpendicular to the direction of loading. It isconcluded that it is possible to detect surface local stress concentration by Rayleighwave acoustoelasticity in polymer components such as those of PMMA.

Acknowledgements

This work was fully supported by the National Natural Science Foundation ofChina and partly by Doctoral Foundation of the National Education Ministry ofChina. We are grateful to Professor Bao Yuhan in China Aviation Industry Com-pany for preparing samples, Professor Fu Yanbin in China Institute of Metrologyin aid of polymer confection and Professor Wang Xingli in Institute of SolidMechanics of Zhejiang University for technical assistance in ultrasonic experiments.

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