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THE I.F. IN METALS AS A FUNCTION OF PHASE VELOCITY OF ELASTIC WAVES
A. Capecchi
To cite this version:
A. Capecchi. THE I.F. IN METALS AS A FUNCTION OF PHASE VELOCITY OF ELASTIC WAVES. Journal de Physique Colloques, 1987, 48 (C8), pp.C8-305-C8-310.
�10.1051/jphyscol:1987844�. �jpa-00227148�
JOURNAL DE PHYSIQUE
Colloque C8, supplbment a u n012, Tome 48, decembre 1987
THE I.F. IN METALS AS A FUNCTION OF PHASE VELOCITY OF ELASTIC WAVES
A. CAPECCHI
Istituto di Scienza delle Costruzioni, Universita di Genova, I-16145 Genova, Italy
ABSTRACT
A great number of results of experiments carried out on some metals at vanishing amplitude and room temperature, reveal that it is possible to plot univocally I.F.
versus the phase velocity of elastic waves. Measures obtained with longitudinal, flexural and torsional waves find their correct place in the diagram, when error sources have been localized and eliminated.
Particular attention is to be paid to the effects of the interaction between specimen motion and suspensions: to this end reference has been made to the concept of oscillators showing nodes characterized by no displacements and rotations (bi- null nodes). When supported at these points, they guarantee that a minimum of energy may escape from specimens.
Bars up to six meters in length, rings, disks up to one meter in diameter have been tested, making it possible to judge the reliability of testing methods and to establish the limits of testing with each type of oscillator. Therefore, plotted against phase velocity, the I.F. gives rise to a trilateral diagram that shows no peaks revealing an experimental behaviour different in shape and absolute values from that predicted from Zener's thermo-elastic relaxation theory.
Torsion tests on rectangular cross section bars, whose velocities are lower than that of distorsional waves, exhibit the same I.F. values measured in flexure in the same velocity field.
1. FOREWORD
In a previous work [ I ] , a dependence of I.F. on the phase velocity c of elastic waves was envisaged in metals at room temperature and vanishing amplitude; for a better understanding of the matter, it may be useful to recall some considerations about wave propagation.
In an infinite, ideal medium, only dilatational and distortional waves exist and they have corresponding velocities c, and c,, independent from frequency f and wavelength A; therefore their relative phase and group velocities are equal.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987844
C8-306 JOURNAL D E PHYSIQUE
When considering standing waves in a bar, dilatational waves change into extensional waves and the distortional waves change into flexural or torsional waves depending on the type of the load applied. Generally, it may be said that all kind of waves in a bar suffer dispersion, that is a variation of velocity c with the wavelength or the cross section sizes of the bar.
Fig.1 shows from KOLSKY 121, the curves of dispersion of c in a cylindrical bar of radius a, versus the ratio a/A; only the first torsional mode shows a constant value that equals c,. The extensional waves reach c,, bar velocity, corresponding to an infinite wavelength, while the flexural wave velocity, starting from zero, shows an initial straight zone and bends to reach c, that is the velocity of ~a~lel'gh Waves near to the final value of the extensional waves. The relations between Young's modulus E and shear modulus G with phase velocity of waves are based upon
E = p c O 2 where p is the density 1.1
G = p c,' 1.2
Phase velocity is defined as
where w is the angular frequency and k = 2r/A is the wavenumber in standing waves. Experimentally, the dispersion curves are confirmed but, for longitudinal waves, beyond a given value of a/A, higher vibration modes appear; therefore it is better to not work far fromcc,.
The flexural wave velocity may be expressed, in the linear initial zone, by
c = k c . r = J ~ . Fig. 1
where r is the radius of gyration of the cross section of the bar around an axis, in the neutral surface, perpendicular to the axis of the bar.
When plotting the diagram of I.F. versus phase velocity of waves, for instance considering Aluminium alloys, the field from 1 to 1000 m/s may be covered by flexural waves, while only one point corresponds to the velocity of torsional waves and another point to longitudinal waves, whatever is the exciting frequency. To fill the gap between flexural and torsional waves velocities, probably for the first time,rectangular cross section bars have been tested in torsion with the purpose of obtaining velocity values lower than c,. The relation
can be obtained from the motion equation where J, is the polar inertia momentum, Jd the torsion modulus i.e. the geometric contribution to the torsional rigidity; if the lengths of the cross-section sides are b > 2s, the approximate formulB
may be used and it is therefore possible to have velocities as low as 250 m/s starting from c2 z 3000 m/s.
An interesting consideration has to be outlined on the suspension points of the oscillators employed; with flexural waves in a bar, the nodes only ensure that displacements are zero, while rotations exist causing interaction with suspension devices. Adopting two dimension oscillators, having only two or more nodal axes, in the cross point of axes, displacements and rotations are null permitting a minimum of energy to escape; a similar node may be called a bi-null node and the development and the use of these oscillators seems to be very interesting for the
increasing reliability of the I.F. measure. From this point of view the most cormnun shapes of oscillators are listed below: all types of pendulums, cantilevers, free- free bars suspended in two points, free-free bars vertically suspended in only one nodal point, rings and, finally, disks resting on center together with other bi-null oscillators. The reasons for this sequence are clear when one examines the forces between specimen and frame into the fixing point or the motion of the specimen in the suspending points. It may be that, in spite of the above, some measures obtained with an unreliable methodology are affected by a still acceptable error, but the difficulty arises when it is necessary to assess what it amount to.
Plotting I.F. versus phase velocity appears to be the first ordering system able to collect all measures of whatever specimen cross sizes, vibration frequencies and kind of exciting waves in a single diagram. (Fig.2) To close this introduction it may be recalled that the phase velocity in rings may be calculated by 1.3, but in disks with 1.7
c 0
c = k S =
\/T --
6 2 4 5 7 5 mJF7
where s is the thickness I 2 1 I nlo 2 1 a ~~~l t I I 8 , 0 1 I 1 I al14
and v the Poisson ratio.
PHASE VELOCITY c ( m/r )
Fig.2 2. EXPERIMENTALS
All tests have been carried out measuring the free decay of natural vibrations of specimen in vacuum; both exciting and revealing has been obtained with no-contact systems working on eddy currents that permit a not critical distance of few millimeters.
In particular, the exciter is formed by winding two coils on a soft iron core; the first for obtaining the magnetic permanent field and the second directly connected to a function generator. Switching off both the supplies to coils, no energy absorption should exist but, for specimens with a very high deformability, it is better to mechanically remove the exciter from the proximity of the specimen because of the residual magnetism. For the revealing system it would be better to use only optical transducers, always with the aim of minimizing the loss of energy. The writer tried to adopt a capacitor system also but it needs a minimum distance between specimen and capacitor plate, together with a perfect parallelism; for obtaining this, the specimen has to be suspended in a stable manner that then increases the energy escape. The measures have been obtained through a digital analyser DATA 6000. The specimen sizes reached about five meters in length for bars and one meter in diameter for disks. It has to be noted that a sufficient mass of specimen is a guarantee against errors, especially working on bars, while binull disks of small diameter are still usable.
It is necessary to recall that, among several measurements on the same sample, the best will reveal the lowest value of I.F. because any possible anomaly will always increase the measure for the losses of energy outside the specimen.
Since the total variation of I.F. measured by the writer on metals may overcome more than two logarithmic decades, in view of the fact that commercial grade metals were employed, for each metal a band of results was accepted; as one can expects, this band is more narrow for the electrolytic copper than for aluminium alloys. The writer obtained I.F. trilateral diagram similar to that shown also for brass, iron and tantalum; with this last metal, onlyfewpoints were obtained because of the difficulty of preparing great-size specimens. However, in this paper only Al-alloys diagram has been shown owing to the granted space.
JOURNAL DE PHYSIQUE
3 . LIMITS OF EACH TYPE OF OSCILLATORS
3.1. DISKS
The disk appears the best and easiest vibrating structure to test in bending if one does -not desire phase velocity values nor lower than 20 m/s; the limii is due to the difficulty of obtaining flatness when diameters reach about one meter and thickness is one millimiter or 1ess.The upper limit is reached,speaking of Al-alloys, when the ratio between diameter and thickness decreases under about twenty. Probably, shear effects are no longer negligible beyond this value.
The disks have to be tested resting on a tip point into the center both horizontally or vertically; the I.F. measures increase up to one logarithmic decade when the disks are supported on three points on a nodal ring even if the tips reach the neutral plane
.
3 . 2 . RINGS
When excited in flexure in its plane, suspended by a thin wire, a ring autocenters a node in the suspension point, then annulling the error in localizing the node, but a rotation is present there.
These oscillators place themselves therefore between disks and bars, regarding the amount of error. Rings are difficult to build and then they are interesting only for demonstrating that a residual error is shown because of interaction with suspension.
3 . 3 . FREE-FREE BARS
Difficulty arises in pin-pointing nodes with the three types of waves; furthermore, in bending, rotation occurs in nodes.
If the bar is placed horizontally, two suspensions are necessary; it is much better to place the specimen vertically, suspended by a thin wire through a hole drilled in one of the nodes. With two suspension, a considerable increase of I.F. is revealed shortening wavelength under two meters.
To excite and reveal torsional vibrations, transducers of the type developed by HANSTOCK & MURRAY [ 3 ] were employed, placed outside a vacuum pipe in which the specimen is suspended.
In exciting flexural waves, a behaviour, judged anomalous by the writer, appears in the region of the upper knee of the diagram; bars show a bell shaped increase of I.F. Many authors found similar curves in several metals, therefore claiming this as a confirmation of Zener's theory of thermoelastic transverse currents.
3 . 4 . CANTILEVERS IN BENDING AND PENDULUMS
These oscillators surely permit a lot of the specimen energy to escape into the fixing frame; for this reason they are to be avoided even if some results appear acceptable.
.5. CROSS SHAPED SPECIMENS
With the purpose of designing a bi-null oscillator formed by beams, a cross shaped specimen has been choosen; from a thin sheet of metal, a cross is cut showing four perpendicular branches. One pair of opposite branches will bend in one direction, the other pair in the other. In this manner a dynamic equilibrium exists in the center point of the cross.
To make these specimens, a machine tool has been prepared composed of a rotary- table on which the specimen is fixed and a compound table supporting a high- velocity hand-operated miller.
A similar oscillator has to be excited and revealed on the four ends of branches, connecting together outputs to obtain a mean value to reduce the quite unavoidable modulation.
3 . 6 . OSCILLATORS WITH ADDED MASSES
Adding of concentrated masses to bars in flexure, as made by BERRY [ 4 ] , appear to reduce I.F. value in respect of specimens that show only distributed mass and vibrating on the same frequency and/or the same velocity. On the contrary
the adding of masses to bars in torsion seems to not alter the measured I.F.
values.
4 . COMMENTS ON RESULTS
In this research there are two "key points": the first aopears to be the singleness
- - .
* -of the I.F. value for round section bars in torsion for whatever frequency and the second is represented by the fact that, in flexure, only bars show a bell shaped behaviour similar to the predictions of Zener's theory. On the contrary the singleness of the I.F. value for extensional waves appears to be less traumatic.
As regards the first claim, the writer found results judged reliable in other authors'papers; for Al-alloys, fig. 3 shows results of FROMMER & MURRAY 151 on free- free bars, COTTZL & ENTWISTLE [6] on fixed-free bars with added masses and on free- free bars with masses, KIMURA-SUZUKI-HIRAKAWA [7] on free-free bars with masses. As it is possible to note, all I.F. values range between 2 and 5.10-6 confirming a quite constant value of I.F. on a field of four logarithmic decades of frequency.
The anomalous increase shown by bars near to the upper Q 1 8
knee of the diagram also appears of the greatest importance;to study this phg = nomenon, tests have been
'
carried out on bars, disks and crosses of the same thickness.& I
To compare the experimental 2
'
results with those predicted
2 '
by Zener's theory, these va- p lues are shown in f i g . 4
'
versus frequency. i i'
No peak is revealed by binull 1 2 a 6 I 2 4 H 4
,
8 2oscillators and moreover, in
the Zener's peak zone, the Fig.3 FREQUENCY C I S correct values are lower by about 60%.
Therefore more facts appear not to confirm Zener's theory:
-
to the left of peak, measured values are much higher than those predicted and they show a quite constant value against the predicted 45' slope decrease-
no peak is revealed by reliable oscillators- the decreasing of values on the right of the peak has a different slope
-
the values obtained by testing in torsion rectangular cross-section bars perfectly enter into diagram 1.F.-phase velocity confirming that there is a continuity of behaviour between extensional, torsional and flexural waves.-
the constant value in the high velocity field.In conclusion, if Zener's theory on transverse thermal diffusion is not confirmed by reliable
experimental measure- ments, two hypotheses may be formulated: the first based upon a dif ferent calculation of the absorption due to the thermal relaxation while the second would detect a different mechanism as responsi- ble for the behaviour revealed. To this end, the writer would like to test disks with thermal transverse bar-
r ier
.
FREQUENCY C I SFig. 4
C8-310 JOURNAL DE PHYSIQUE
SPECIMEN n"
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
(I) (I) (I) (I) (I) (I)
Expérimental OSCILLATOS and MODE cantilever cantilever cantilever cantilever free-free beam free-free beam free-free beam free-free beam free-free beam free-free beam
results VELOCITY
m/s 3,65 4,23 5,0 6,12 7,8 21,5 24,8 30,3 36,4 45,1 (Ill)free-free beam 100,0 (I) free-free beam 104,0 (III)free-free beam 125,0 (I)
(I) (I)
free-free beam free-free beam
123,0 31,3 free-free beam 227,0 (Ill)free-free beam 243,0 (Ill)free-free beam 518,0 (I)
(I)
free-free beam free-free beam
disk disk disk disk disk disk disk disk disk (II) ring (IV) ring (II) ring (IV) ring (I) ring (V)torsion bar (V)torsion bar (V)torsion bar (Ill)torsion bar '.
(Il)extensional bar '.
(Df (Df
ree-free beam ree-free beam
671,0 958,0 16,5 26,0 16,0 22,7 100,0 217,0 228,0 243,0 355,0 190,0 130,0 239,0 531,0 558,0 240,0 418,0 941,0 3138,0 5009,0 4,83 5,1
obtained on Al FREQUENCY
c/s 0,773 0,985 1,383 2,066 3,55 2,54 3,39 5,06 7,29 11,22 59,9 23,9 86,5 86,7 13,5 56,7 129,7 302,0 2481,6 2023,0 13,11 34,53 50,15 104,23 1047,7 1649,0 1347,0 842,1 4347,9 1285,0 373,0 630,0 3365,0 1914,0 209,4 1763,6 3921,0 18079,9 2023,0 1,28 1,44
-alloy spec:
Q"1
7,3x10""
7,0x10""
6,9x10 "
7,0x10 * 5,7x10""
7,0x10""
5,2x10""
4,5x10""
3,0x10""
1,7x10""
8,0x10 5
6,3xl0"5
6,0x10 s
7,2xl0"5
4,4x10 "
2,7x10 5
3,16x10""
1,9x10 5
1,0x10 5
1,5x10 5
7,6x10""
7,3xl0"5
4,0x10""
5,4x10""
7,0xl0"5
4,6x10"5
4,0xl0"5
7,0x10"5
3,0xl0"5
6,7xl0"5
8,5xl0~5
4,9x10 5
2,8x10"5
2,3xl0"5
4,8x10 5
3,lxl0-5
I,7xl0-5
l,15xl0~5
4,9x10""
7,34x10 "
7,8 xlO "
imens
SIZES mm 2x30x1500 2x30x1500 2x30x1100 2x30x 900 2x20x1760 20x20x6001 20x20x5500 20x20x4500 20x20x3750 20x20x3000 20x100x3013 50x20x3242 20x20x2535 20x20x1090 8x20x1744 100x20x3013 50x20x3242 100x20x3013 20x20x200 50x20x243
<j> 994x2
$ 560x2
* 215x0,5
<t> 150x0,5
<|> 100x2 4> 97x3
* 125x4
* 200x6,8
* 60x3
* 94,2x4,6x20
* 181,6x5x12
<t> 198x10x12
• 198x10x12
$ 181x26x12 2x50x575 2x30x600 3x20x600
<)> 15x260 20x20x2479 2x50x2750 2x20x2750
5. REFERENCES
[1] Capecchi A., Capurro M., J. de Phys. Coll. C9, Suppl. 12, 44j, 447, (1983) [2] Kolsky H., Stress waves in solids, Clarendon Press, Oxford, (1953).
[3] Hanstock R.F., MurrayA., J. Inst. Met., 72, 97, (1947).
[4] Berry B.S., J. Appl. Phys., 26, 884 and 1221, (1955).
[5] Frommer L., Murray A., J. Inst. Met., 70, 1, (1944).
[6] Cottel G.A., Entwistle K., J. Inst. Met., 74, 373 (1948).
[7] Kimura S., Suzuki T., Hirakawa H., Phys. Lett., 81A, 5, 302, (1981).
