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ULTRASONIC ATTENUATION IN METALS
DEFORMING AT HIGH RATES OF STRAIN : DATA ANALYSIS BY USING A KINETIC MODEL OF
DISLOCATION LOOPS
J. Shioiri, K. Sakino
To cite this version:
J. Shioiri, K. Sakino. ULTRASONIC ATTENUATION IN METALS DEFORMING AT HIGH RATES OF STRAIN : DATA ANALYSIS BY USING A KINETIC MODEL OF DISLOCATION LOOPS.
Journal de Physique Colloques, 1987, 48 (C8), pp.C8-197-C8-202. �10.1051/jphyscol:1987827�. �jpa-
00227131�
ULTRASONIC ATTENUATION IN METALS DEFORMING AT HIGH RATES OF STRAIN : DATA ANALYSIS BY USING A KINETIC MODEL OF DISLOCATION LOOPS
J. SHIOIRI and K. SAKINO
College of Engineering, Hosei University, Koganei, Tokyo, Japan
Abstract
-
In order to obtain experimental knowledge on the behaviour of dis- locations at high rates of strain, time-resolved measurements of the ultra- sonic attenuation and velocity in specimens undergoing dynamic plastic defor- mation have been tried. In this paper, some analyses of the ultrasonic data for polycrystalline aluminium are made by using a kinetic model for the motion of dislocations cutting through the forest dislocations. The model given its quantitative basis by the ultrasonic data can describe fairly well the strain rate dependency of the flow stress over a wide range of strain rate.I
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INTRODUCTIONExperimental methods of obtaining information on the behaviour of dislocations at high rates of strain have been mostly by measurements of the flow stress and obser- vations of the dislocation structure after deformation. In order to develop a more direct method, the present authors group tried time-resolved measurements of the ultrasonic attenuation a!?d velocity in specimens undergoing dynamic plastic defor- mation /I-6/. Recently, by improving the time-resolution capability of the ultra- sonic apparatus, measurements at strain rates as high as about 8000 /sec have become possible / 3 / . However, the kinetic model of dislocations which has been used for drawing the experimental information from the ultrasonic data is rather unsatisfacto- ry, and refinement has been felt necessary.
In the present work, a model is used newly in which the elastic flexibility of the glide dislocations, which is expected to play an important role when they pass through the point obstacles, is taken into account, and ultrasonic data for high purity poly- crystalline aluminium are analysed. Further, by using the obtained quantitative in- formation on dislocations, the flow stress is calculated and compared with the direct- ly measured flow stress.
I1
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ULTRASONIC ATTENUATION DUE TO DISLOCATIONS UNDER GLIDE MOTIONOn the basis of the work by Frost and Ashby / 7 / , Klahn et al. /8/ presented a mathe- matical expression for the motion of glide dislocations passing through point obsta- cles. In their model, as shown in Fig. 1 , a square array of point obstacles and the bowins of the glide dislocations due to the elastic flexibility were taken into
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987827
JOURNAL D'E PHYSIQUE
POINT
/--- -1.r
OBSTACLESFig. 1 - Dislocations passing through a square array of point obstacles.
t=O
---1) X
account. Further, it was assumed that two rate controlling mechanisms, i.e. the thermally assisted cutting of the point obstacles and the viscous drag against run- ning dislocations, are simultaneously operative and accordingly the motion of the glide dislocations is jerky. Their model is outlined in the following.
As shown in Fig. 1, a glide dislocation comes into contact with a row of point obsta- cles at t=O, bows out and cuts the obstacles with the aid of the thermal activation at t=tl, and then continues viscous drag-controlled jump motion and again contacts the next row at t=t3. By denoting the interval of the jump by t2, the mean velocity of the glide dislocation is geven by
where, L is the distance between the obstacles. Assuming the slopes of the glide dis- locations are small compared with unity, the configuration of the dislocation at t<ti is given by
where T is the resolved shear stress, b is the Burgers vector,
r
is the line tension of the dislocation, tret is the time constant of retardation, B is the damping con- stant. Further, assuming the rectangular force barrier of width d, tl is given by1 = JO1v exp[-(2rd/kT) (sin t - sin 8)1 dt (3) where v is the frequency factor, 8 is the slope of the bowed dislocation at the con- tact points with the obstacles, and Bc is the critical value of 8 at which cutting occurs athermally. The free jumping interval t2 is given as
where
7
is the average value of y at t=tl. By using the values of tl and t2 deter- mined by Eqs. (3) and (&), respectively, Eq. (1) gives the shear strain rate aswhere NL is the total length of the glide dislocations per unit volume.
Besides the elastic flexibility, the inertial effect may be an important factor for cutting.the obstacles. However, except under special conditions such as the super- conducting state, the damping constant B has a sufficient magnitude for causing an overdamping condition, and accordingly the inertial effect will be negligible even at high strain rates.
if the period of the jerky motion of the glide dislocations, tl+t2, is sufficiently small compared with the period of the ultrasonic wave, l/f (f is the ultrasonic fre- quency), the effects of the gliding dislocations on the attenuation and velocity are given by using j of Eq. (5) as
(Ah), = ~G(df/d~ )T=T0/2f ( 6 ) and (AV/V), = o[(AX),~I ( 7 ) respectively, where 52 is the orientation factor, G is the elastic modulus in shear, To is the stress imposed to cause the dynamic plastic deformation. Further, at lower
strain rates, where the waiting time for the thermal activation at obstacles is the dominant rate controlling factor, t2,tret<<t
,
and by neglecting the high frequency(ultrasonic frequency) changes in L and NL, is given in the following form:
Since, as seen in Eq. ( 7 ) , (AV/V), is a small quantity one order higher than (AX),, it is difficult to use it in the analysis of the ultrasonic data simultaneously wlth
(Ax),. On the other hand, as shown in Section IV, Eq. (8) plays an important role in place of Eq. (7).
I11 - EXPERIMENTAL METHOD
In the present experiment, an ultrasonic apparatus with high time resolution capabil- ity is required. The devised ultrasonic apparatus and set-up around the specimen are shown in Fig. 2. The ultrasonic apparatus has two phase detectors and to the second one the reference wave is fed after phase shift by n/2. The output pulses of these two phase detectors, A1 and A2, give the instantaneous values of the ultrasonic atten- uation and velocity. The resolution time of the apparatus is 3 Usec for repeated pulse operation and 1 usec for isolated pulse operation. The ultrasonic pulses are sent at right angles to the direction of the dynamic compression imposed by the split Hopkinson pressure bar apparatus. In the present work, interest is in the attenuation due to dislocations under glide motion, and, therefore, it is necessary to separate the attenuation due to the gliding dislocations from the attenuation due to the other sources such as the pinned dislocations. Since the attenuation due to the dislocation is very large when it is under the glide motion compared with when it is in the pinned down condition, the attenuation due to the gliding dislocations under dynamic defor- mation can be determined from the drop of attenuation observed when the deformation
Transmitting transducer
Specimen
n12 phase shifter
transducer Oscilloscope
1
(1 ) Specimen (2) Transducers (3) Hopkinson bars
1 Transient
digital memory * - Phase detector 2
a=6 mm: b=c=5 mm:
d=e=3 mm: f=10 mm.
Fig. 2 - Ultrasonic apparatus (left) and set-up around specimen (right).
JOURNAL DE PHYSIQUE
-
time
Fig. 3 - Changes in the output pulse heights, Aland A2, due to deformation.
[Left] Repeated pulse operation: aluminium, strain rate 70 /see, 17x1 7x1 7 mm3 speci- men, 30 MHz.
[~ight] Isolated pulse operation: aluminium, strain rate 4000 /see, 6 ~ 5 x 5 mm3 speci- men, 10 MHz.
stops suddenly. At very high strain rates (>2000 /see) the isolated pulse method was used: i.e. three isolated pulses sent before, at the end of and after the deformation can give necessary information. In Fig. 3, some examples of the records of the two output pulses, A1 and A 2 , before, during and after the deformation are shown. De- tails of the experimental technique were already reported in Refs. 7 and 6.
IV - EXPERIMENTAL RESULTS AND ANALYSES
Ultrasonic measurements were made for high purity (99.999%) polycrystalline aluminium.
The ultrasonic frequenc was 10 MHz. In Fig. 4 the attenuation due to dislocations under glide motion, (AA~,, determined from the drop of the attenuation at the end of the deformation is plotted against the strain rate i . The change in the ultrasonic velocity observed when the deformation suddenly stopped was negligibly small as was expected from Eq. (7).
0 Experiment
I I ,
Fig. 4 - Attenuation due to dislocations under glide motion vs strain rate:
0 Experiment.
-
Fitted theoretical curve.0 2 4 6 8 10
S t r a i n Rate ( s e c - l ) x103
sonic data of Fig. 4 were analysed by using Eqs. (6) and (8). For the physical quan- tities and constants the following values and relationships were used: that is, G=
0.267~1012 dyn/cm2; b=2.8~10-~ cm; ~=2.5~10-4 dyn sec/cm2 / 9 / ; r = ~ b ~ / 2 ; Q=O.OL /lo/;
€=y/2 and 0 ~ 2 ~ . Further assumptions d=b and sin 8,=0.2 give ~ b 3 / 5 for the activation energy of cutting the forest dislocations. This value is equal to the value proposed by Ashby /I I/. For the frequency factor v the Debye frequency vD=1013 /see was used;
very often v=uDb/L is used, but the trial motion against the potential barrier may not be the oscillation of the dislocation loop which is strongly damped but a more localized oscillation of the atoms around the intersection point. Equation (8) gives
~=0.475~10-5 cm. Further, assuming that L and NL are independent of k , and using the above obtained value of L, fitting of Eq. (6) to the experimental data of Fig. 4 was tried. The best fit was obtained when NL=2.€%107 /cm2. The fitted curve is shown in Fig. 4. The dislocations contributing to the deformation are rather minor.
Using Eqs. (2 to 5) with the above values of L and NL, the flow stress was calculated.
Results are shown in Fig. 5. The open circles in Fig. 5 are the experimental values simultaneously measured with the ultrasonic measurements. The calculated curve shows a remarkable change in the strain rate sensitivity of the flow stress at E around 10000 /sec. In the process of the calculation, it was also noticed that the change occurs when tl becomes negligibly small compared with t2, and this implies that the change is due to the transition in the rate controlling mechanism of the dislocation motion. However, it must be noted that this change appeared in the extrapolation region of the ultrasonic measurements and also the flow stress measurements simulta- neously made, and therefore, in order to clarify the problem, measurements at strain rates above 10000 /sec are needed. Since, at present, the ultrasonic measurements in this strain rate range are quite difficult, the direct measurements of the flow stress have been made. In order to avoid the inherent difficulties in applying the split Hopkinson pressure bar method to very high strain rates, very small speci- mens (2 mm both in diameter and length) were used.' The material of the specimens was the same as in the foregoing ultrasonic measurements. Results are plotted in Fig. 5 with full circles. The results show a steep change in the strain rate sensitivity very similar to the calculated curve, and it may be concluded that the kinetic dislo- cation model quantitatively reinforced by the ultrasonic data simulates fairly well
Fig. 5
-
Flow stress vs strain rate.-
Calculated on the basis of the ultrasonic data.0 Measured simultaneously with the ultrasonic measurements.
Measured by using small specimens.
7 0 - - 6 0 -
2
z 5 0 - D 40-
2
v) 30- U V]20
K! 10
0
Strain Rate
6 (s~c-'1
~ = 0 . 0 5 288 K
-
Calculation-
0 Experiment (Simultaneous)
-
Experiment (Small Specimen)I I 1 1 1 1 1 1 I I 1 l 1 l t l l I
100 1000 10000
JOURNAL DE PHYSIQUE
the strain rate dependency of the flow stress over a wide range of strain rate in- cluding the thermal activation flow and the viscous drag-controlled flow. However, it must be noted that, in the present calculation, the Schmid factor was taken at 0.5. This was clearly an oversimplification. Furthermore, in the present analysis, the model presented by Klahn et al., in which a square array of point obstacles was assumed, was used, but the randomness in the array of the obstacles should play an important role.
ACKNOWLEDGEMENTS
This work is a part of the project "Research on Ultrasonic Spectroscopy and Its Applications to Materials Science" aided by Grant-in-Aid for Special Project Re- search of The Ministry of Education, Science and Culture of Japan. The authors are also grateful to The Mitsubishi Foundation for their financial support.
REFERENCES
/I/ Shioiri, J. and Satoh, K., Inst. Phys. Conf. Ser. No. 21 (1974) 154.
/2/ Shioiri, J. and Satoh, K., ibid. No. 47 (1980) 121 /3/ Shioiri, J. and Satoh, K., ibid. No. 70 (1984) 89.
/ 4 / Shioiri, J. and Satoh, K., Journal de Physique, Colloque C5, supplement au n08
Tome 46 (1985) C5-3.
/5/ Shioiri, J., Sakino, K. and Satoh, K., ibid. Colloque C10, supplement au n012, Tome 46 (1985) C10-333.
/6/ Shioiri, J., Satoh, K. and Sakino, K., Review of Progress in Quantitative Non- destructive Evaluation, Vol. 5B, Plenum, (1986) 1577.
/7/ Frost, H. 5. and Ashby, M. F., Tech. Reps. No. 1, Div. Eng. and Appl. Phys., Harvard University, (1970).
/8/ Klahn, D., Mukherjee, A. K. and Dorn, J. E., Proc. 2nd Int. Conf. on The Strength of Metals and Alloys, American Soc. Metals, (1970) 951.
/9/ Gorman, J. A., Wood, D. S. and Vreeland, T. Jr., J. Appl. Phys. @ (1969) 833.
/lo/ Hikata, A., Truell, R., Granato, A., Chick, B. and Lucke, K., J. Appl. Phys.
27 (1969) 396.
/11/ Ashby, M. F. and Frost, H. J., Constitutive Equations in Plasticity, MIT Press, (1975) 117.
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