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CALCULATION OF THERMOELASTIC BENDINGS OF THIN PLATES APPLICATION TO THERMAL
DIFFUSIVITIES MEASUREMENTS
G. Rousset, F. Charbonnier, F. Lepoutre
To cite this version:
G. Rousset, F. Charbonnier, F. Lepoutre. CALCULATION OF THERMOELASTIC BENDINGS OF
THIN PLATES APPLICATION TO THERMAL DIFFUSIVITIES MEASUREMENTS. Journal de
Physique Colloques, 1983, 44 (C6), pp.C6-39-C6-42. �10.1051/jphyscol:1983606�. �jpa-00223165�
JOURNAL D E PHYSIQUE
Colloque C6, supplement au nOIO, Tome 44, octobre 1983 page C6- 39
CALCULATION OF THERMOELASTIC BENDINGS OF T H I N PLATES A P P L I C A T I O N TO THERMAL D I F F U S I V I T I E S MEASUREMENTS
* +
G. Rousset, F. ~harbonnier* and F. Lepoutre
EcoZe PoZytechnique, De'partement de Ge'nie Physique, CP 6079 A, Montre'aZ H3C 3 A 7 , Canada
+C.N.R. S . G. R. 14, EcoZe CentraZe de Paris, 92290 Chatenay-MaZabry, France
A+ ~ a b o r a t o i r e d 'Optique .Physique, ESPCI, 20, rue VauqueZin, 75231 Paris Cedex 05, France
RQsume - Le calcul des d6formations thermoelastiques d'une plaque fine montre qu'aux frQquences &levees, celles-ci peuvent perturber fortement l'effet photoacoustique. Une application de ce calcul B la mesure des diffusivitbs est pr6sentee.
Abstract - The calculation of the thermoelastic bending of thin plates shows that this effec-t can dominate the photoacoustic effect with thermally thick samples. An application of this calculation to thermal diffusivity measure- ments is given.
Several papers have recently reported observations, in photoacoustic experiments, of strong signals due to the thermoelastic deformations of the sample / I / . Further- more, these deformations are directly used in piezoelectric-photoacoustic detec- tion / 2 / . In our case we have been concerned with this problem when we have used photoacoustic detection to measure thermal diffusivities / 3 / .
In a first part, we present a calculation explaining quantitatively the signals observed in the particular case of thin plates. We call 7 the displacement
(components ur, uO, uz) (Fig.
1 ) .Fig. 1 - Axes and notations used in the calculation.
The basic equations / 4 / / 5 / which couple 7 and TS (temperature in the sample) are simplified with the following hypotheses (a) no O dependence and uo
=0
(cylindrical symmetry), (b) plane stresses
(R>> Ls and Ts independent of r), (c) no coupling terms with in the heat diffusion equation, (d) body forces
'neglected, (e) inertial forces neglected, (f) heat losses from the sample to the surrounding gas neglected. All these hypotheses are easily satisfied, except for (b) : Ts independent of r can only be approximated, in practice, by a uniform illumination of the almost totality of the sample surface. With these hypotheses, the projections of the coupled equations are
:Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983606
JOURNAL DE PHYSIQUE
where v is the Poisson's ratio, a~ the linear thermal expansion coefficient,
cisthe thermal diffusivity, f the modulation frequency. We have solved the system (1) to (4) with the boundary conditions of a simply supported plate at r
=K (Fig. 1).
The solution for ur gives the signal obtained when the piezo-electric photoacoustic detection is used and has been discussed elsewhere / 5 / . The solution for uz gives the signal due to the bending of the plate
:where I . is the power absorbed by the sample, A s the thermal conductivity of the sample and as
=(~+j)(IIf/a~)~~~. Eq. (5) shows that the thermoelastic bending
2 3
is important for thin and large plates (R /Ls large), made of a strongly expansi- ble material.
If the sample consitutes one of the walls of a photoacoustic cell, we can calculate the pressure P induced by the thermoelastic bending using the mechanical piston model of McDonald and Wetsel 161 :
where y is the heat capacities ratio of the gas, and P and Vo the pressure and the volume of the gas.
The frequency dependence of such a bending effect is very interesting to point out
;let us note f
=a / & 2 the characteristic frequency of the sample. When f < fc
C
s s
the thermoelastic pressure (eq. 6) is independent of f, when f > fc it decreases
as f-1. Fig. 2 shows the comparison of this frequency dependence (curve @ )
with the ones of the photoacoustic pressure (curve 0 ) and of the acoustic
piston of McDonald and Wetsel /6/ (free thermal expansion) (curve @ ),when the
pressure signals are detected on the side of the non illuminated surface of the
sample / 3 / . It appears clearly that the thermoelastic effect becomes several
orders of magnitude larger than the photoacoustic effect at high frequency. When
the signals are detected on the side of the illuminated sample surface, the
photoacoustic pressure decreases much slowly (as £-I or f-312) / 7 / and remains
of the same order of magnitude as the thermoelastic pressure.
Fig. 2 - Theoretical amplitude of the three effects due to : a). Thermal diffusion- 0. Free thermal expansion of the sample - 0 . Thermoelastic bending of the sample.
The chosen sample is here a thin plate of ~n(1 =.5mm, ~ = 1 2 m ) .
In the second part of this poster, we present an application of this calculation to the measurement of thermal diffusivities. The experiment is described in figure 3. The sample is placed between two rigid annular knife edges which autho- rize the displacements along its surfaces but forbid the ones along its normal.
Thus the sample is simply supported. The disc is illuminated as uniformely as possible by a modulated beam. The modulation frequency f is adjustable from some Hz to some kHz. A probe beam is reflected on one of the surfaces of the sample.
When the plate bends, the normal of its surface rotates of
Oand the reflected beam rotates of 2 0 . The reflected probe beam is then detected by a sensor position.
The signal, monitored by a lock-in amplifier, is recorded as a function of the frequency. In order to prevent acoustic resonances, the maximum frequency must be much smaller than C/R (C being the sound velocity in the sample). Thus 2 0 equals
- 1
R .uZ(o, ls/2).
Fig. 3 - Experimental set-up for the measurement of sample thermoelastic bending.
@ The sample @ is mounted between two r i g ~ d annular knife- edges @ ; Xenon arc -
a Mecnan~cal chopper -
@ Lens - a He-Ne laser
(probe beam) - @ Position
sensor.
JOURNAL DE PHYSIQUE
The figure 4 shows the comparison between experiment and theory for three metals.
The experimental points of Figure 4 lead to fc and thus to a,. The agreement with the data of other methods is good /8/ and the accuracy is about 5 %.
Fig. 4
Experimental thermoelastic bending
(0
:stainless steel 304 L,
05 t2$z A nickel, X : zinc) (Xs
=.5mm,
R
=12mm). The theoretical values are given by the continuous curves.
One finds for a : '-6 2 -1 0 : a = 3 . 7 1 0 m s ,
-5 2 -1 A :
CL= 2 . 2 10 m s ,
x : as
=4.1 m2 s-I.
REFERENCES
/I/ PELZL J. and BEIN B.K., XVth Bunsen Kolloquium, Photo Akoustic Spectroscopic und Ihre Andwendungen, Dusseldorf, FRG 1982.
CAHEN D., Workshop on Photoacoustics, Bad Honnef FRG 1981.
/ 2 / JACKSON W. and APER N.M., J. Appl. Phys. 51 (1980) 3343.
131 CHARPENTIER P., LEPOUTRE F. and BERTRAND L., 3. Appl. Phys. 53 (1982) 608.
/4/ NOVACKI W., Thermoelasticity, Pergamon (1962).
/ 5 / ROUSSET G., LEPOUTRE F. and BERTRAND L., to be published in J. Appl. Phys.
/6/ McDONALD F.A. and WETSEL G.C., J. Appl. Phys. 49 (1978) 2313.
/ 7 /
