Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar
The propagation coefficient associated with the wave dispersion and attenuation in viscoelastic bars is calculated experimentally. The phase velocity and the attenuation coefficient in relation to the frequency are obtained from a series of impact tests on a PMMA Hopkinson pressure bar using ball be...
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irk-123456789-1409912018-07-22T01:22:50Z Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar Theodorakopoulos, I.D. The propagation coefficient associated with the wave dispersion and attenuation in viscoelastic bars is calculated experimentally. The phase velocity and the attenuation coefficient in relation to the frequency are obtained from a series of impact tests on a PMMA Hopkinson pressure bar using ball bearing projectiles of different diameters. A wave dispersion correction technique is applied using the Fast Fourier Transforms of the longitudinal strains associated with the incident and reflected waves. Експериментально обчислено коефіцієнт поширення, який має зв'язок з дисперсією хвилі та коефіцієнтом затухання хвилі. Вказані дисперсія та коефіцієнт затухання отримані для матеріалу ПММА в залежності від частоти в серії дослідів з удару за допомогою стержня Гопкінсона. де використані кулі у вигляді сферичних підшипників різного діаметру. Застосовано метод корекції дисперсії хвиль із застосуванням швидкого перетворення Фур‘є щодо поздовжніх деформацій, який зв’язаний з падаючою та відбитою хвилями. 2015 Article Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar / I.D. Theodorakopoulos // Прикладная механика. — 2015. — Т. 51, № 3. — С. 134-144. — Бібліогр.: 9 назв. — англ. 0032-8243 http://dspace.nbuv.gov.ua/handle/123456789/140991 en Прикладная механика Інститут механіки ім. С.П. Тимошенка НАН України |
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The propagation coefficient associated with the wave dispersion and attenuation in viscoelastic bars is calculated experimentally. The phase velocity and the attenuation coefficient in relation to the frequency are obtained from a series of impact tests on a PMMA Hopkinson pressure bar using ball bearing projectiles of different diameters. A wave dispersion correction technique is applied using the Fast Fourier Transforms of the longitudinal strains associated with the incident and reflected waves. |
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Theodorakopoulos, I.D. |
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Theodorakopoulos, I.D. Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar Прикладная механика |
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Theodorakopoulos, I.D. |
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Theodorakopoulos, I.D. |
| title |
Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar |
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Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar |
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Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar |
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Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar |
| title_full_unstemmed |
Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar |
| title_sort |
determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic hopkinson pressure bar |
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Інститут механіки ім. С.П. Тимошенка НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/140991 |
| citation_txt |
Determination of the useful frequency for the calculation of the attenuation coefficient and phase velocity in a viscoelastic Hopkinson Pressure bar / I.D. Theodorakopoulos // Прикладная механика. — 2015. — Т. 51, № 3. — С. 134-144. — Бібліогр.: 9 назв. — англ. |
| series |
Прикладная механика |
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AT theodorakopoulosid determinationoftheusefulfrequencyforthecalculationoftheattenuationcoefficientandphasevelocityinaviscoelastichopkinsonpressurebar |
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2025-07-10T11:41:55Z |
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2025-07-10T11:41:55Z |
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1837260050621530112 |
| fulltext |
2015 ПРИКЛАДНАЯ МЕХАНИКА Том 51, № 3
134 ISSN0032 – 8243. Прикл. механика, 2015, 51, № 3
I . D . T h e o d o r a k o p o u l o s
DETERMINATION OF THE USEFUL FREQUENCY FOR THE CALCULATION
OF THE ATTENUATION COEFFICIENT AND PHASE VELOCITY IN
A VISCOELASTIC HOPKINSON PRESSURE BAR
Manufacturing Technology Division, Department of Mechanical Engineering,
National Technical University of Athens, 9 Iroon Polytechniou, 15780, Athens, Greece.
Email: john_theodorakopoulos@instron.com
Abstract. The propagation coefficient associated with the wave dispersion and attenua-
tion in viscoelastic bars is calculated experimentally. The phase velocity and the attenuation
coefficient in relation to the frequency are obtained from a series of impact tests on a
PMMA Hopkinson pressure bar using ball bearing projectiles of different diameters. A wave
dispersion correction technique is applied using the Fast Fourier Transforms of the longitu-
dinal strains associated with the incident and reflected waves.
Key words: viscoelastic Hopkinson pressure bar, attenuation coefficient, phase veloc-
ity, FFT wave dispersion correction technique.
1. Introduction.
The stress wave propagation in linear elastic or viscoelastic bars is studied in various
experimental techniques, such as the split Hopkinson pressure bar method, which is used to
determine the dynamic/impact behaviour of materials at high strain rates. The Split Hopkin-
son Pressure Bar (SHPB) apparatus was brought to maturity in 1949 by Kolsky [1]. It con-
sists of two long uniform cylindrical pressure bars, known as the incident and transmitter
bars, and a short specimen sandwiched between them. A compressive pulse is produced by
the impact of a striker bar (projectile) at the free end of the input bar. The projectile is fired
at the input bar at various velocities, producing this way a rectangular pulse. When the pulse
reaches the specimen, part of it is reflected back to the input bar, while the remaining pulse
is transmitted to the output bar. The amplitude of the transmitted and reflected waves de-
pends on the impedance mismatch between the specimen and the pressure bars. The three
waves, incident, reflected and transmitted [ ( ), ( ), ( )]I R Tt t t are picked up by a pair of
strain gauges located usually at the mid-point of each bar. The gauges are connected to a
Wheatstone bridge station, and the signals after they have been amplified, they are con-
verted to force-time pulses using a data logger. Since the conventional SHPB apparatus was
designed to employ linear elastic steel bars of high mechanical impedance, the use of speci-
mens made of low density cellular materials would lead to errors in the experimental meas-
urements [2]. A different arrangement of the SHPB system made of a lower impedance
PMMA material was successfully used to minimise these errors. In this current study the
wave attenuation and phase velocity of the pulses produced by the impact of a ball bearing
projectile to a strain gauged PMMA Hopkinson bar are reported.
2. Theoretical considerations.
The conventional SHPB apparatus uses linear elastic pressure bars and the dispersion of
the waves that propagate in the bars will not lead to important errors; therefore, the strains
measured by the gauges can be shifted to the specimen-bar interfaces in order to calculate
135
the three waves [ ( ), ( ), ( )]I R Tt t t and, consequently, to determine the force and particle
velocity at the end of each bar using the following relationships:
( ) [ ( ) ( )]in I RF t E A t t ; (1)
( )out TF E A t ; (2)
[ ( ) ( )]in I RV C t t ; (3)
( )out TV C t , (4)
where, following the notation, A is the cross-sectional area of the bars, E the Young’s
modulus of the material and C the stress wave velocity.
Studies have showed that, in the case of dynamic testing of cellular materials and gener-
ally materials of low mechanical impedance, the classical SHPB method is inaccurate [3, 4].
From equations (1 – 4) it is obvious that, knowledge of the output force and velocity re-
quires only an accurate measurement of the strain imposed to the specimen by the transmit-
ted wave. Additionally, the input force and velocity cannot be determined with any degree
of confidence, if the mechanical impedance of the sample is very small in comparison to the
mechanical impedance of the pressure bars. As a result of this mismatch, the majority of
incident wave is reflected back into the input bar, resulting in a transmitted pulse of very
low magnitude. In this case, the input force tends to zero, since I is almost equal to R .
Thus, for this type of materials, the use of low impedance pressure bars is essential for the
accuracy of the tests.
Usually, the steel bars of the classical SHPB apparatus are replaced by PMMA (or ny-
lon) rods. This material has been used by several researchers in dynamic tests with the
SHPB and is considered to be suitable for this type of applications. The only limitation is
that, due to the viscoelastic properties of the material, the wave dispersion effects are in-
creased. However, a number of methods have been developed to include the dispersion ef-
fects into the dynamic analysis with the SHPB technique and software packages have been
developed, which can perform the dispersion correction in minutes. Tests with PMMA bars
showed an improvement in accuracy of about 200 times of that of the conventional steel
bars [3].
Lifshitz and Leber performed the dispersion correction in the frequency domain after
employing an FFT algorithm by adjusting the phase of each Fourier component [5]. They
indicated that, the accuracy of the dynamic stress-strain curves depends not only on the ac-
curacy of the predicted shapes of each individual pulse but, also, on their positions along the
time axis. A number of tests with the SHPB system showed that, even a small change in the
value of the stress wave velocity could cause a shift of a few microseconds in the position of
the measured pulses. They reported that the main source of the oscillations in the stress-
strain curves after performing the dispersion correction is this shift of the incident, reflected
and transmitted pulses.
Zhao and Gary examined the importance of the dispersion effects in viscoelastic rods
[6]. They suggested that, a three dimensional description of the wave propagation should be
used in order to improve the accuracy of any SHPB system incorporating viscoelastic bars.
The approach for the correction of the wave dispersion was based on the Pochhammer and
Chree wave solution for an infinite elastic bar. Although this solution is only an approxima-
tion for a bar of finite length, it is accepted by many authors and provides accurate results
when it is used with the SHPB system. The Pochhammer and Chree frequency equation in
the case of viscoelastic bars takes the following form:
2 2 2 2 2
1 1 0 1
2
( ) ( ) ( ) ( ) ( ) ( ) ( )f J a J a J a J a
a
2
1 04 ( ) ( ) 0J a J a , (5)
136
where
2
2
( ) 2 ( )
,
2
2 2
( )
, is the complex change in the
phase function, a is the radius of the bar, 0J and 1J are first order Bessel’s functions and
( ) , ( ) are two material coefficients. Note, that, in the above equation is the com-
plex change in the phase function of the angular frequency . The real part of gives the
relationship between the frequency and the phase velocity and the imaginary part gives the
relationship between the frequency and the damping coefficient of the bars.
To illustrate the importance of the dispersion effects they carried out tests with the
SHPB using PMMA rods [6]. This material is described by a number of Voigt or Kelvin
elements connected in series with a spring. The elastic component (spring) and the viscous
component (dashpot) are used to simulate the viscoelastic behaviour of the polymer
(PMMA). In order to determine the spring and damping constants, the waves, at two or
more different points in the bar, were recorded and an identification process, based on an
inverse calculation technique, was applied. Once the constants were identified the rheologi-
cal model was used for the dispersion correction.
Bacon [7] presented an experimental technique based on the method used by Lundberg
and Blanc [8] to determine the propagation coefficient of a viscoelastic rod. This technique
is used to perform SHPB tests with viscoelastic bars by taking into account the wave disper-
sion and attenuation in the bars. According to this method, the propagation coefficient and
the wave attenuation, which are both functions of the frequency, are calculated experimen-
tally. Bacon used the one-dimensional theory of wave propagation in viscoelastic bars and
indicated that the Pochhammer and Chree frequency equation as well as knowledge of the
bar mechanical properties are not necessary.
Bacon also developed a two point strain measurement method for the separation of the
waves propagating in an elastic, or viscoelastic, SHPB taking into account the wave disper-
sion and attenuation in the bars [9]. He performed the wave separation in the frequency and
time domains and showed that the phase velocity and damping coefficient of the bars can be
determined theoretically, as well as experimentally. His technique is a generalization of the
Lundberg and Henchoz method, which is based on the one dimensional elastic wave propa-
gation theory.
As mentioned above, PMMA exhibits viscoelastic behaviour, therefore, the wave at-
tenuation and velocity in the pressure bars are both functions of the frequency.
Consider an axially impacted viscoelastic bar of length L, with a set of strain gauges
mounted at 0x , as schematically illustrated in Fig. 1.
Fig. 1. Axially impacted viscoelastic rod.
According to the one-dimensional theory of wave propagation, the equation of axial
motion in the frequency domain is:
2
2
2
( , ) ( , )x x
x
, (6)
where, ( , )x and ( , )x are the Fourier transforms of the stress and strain, respectively.
The material is linear viscoelastic, thus, the constitutive relation between stress and strain is:
( , ) ( ) ( , )x E x , (7)
0x x d
Ball bearing projectile Positive direction
137
( )E is the complex Young’s modulus of the material. The coefficient of the wave propa-
gation, which is representative of the wave dispersion and attenuation in the viscoelastic bar,
is defined as:
2
2
E
. (8)
Using relationships (7) and (8) the wave equation becomes:
2
2
2
( , ) 0x
x
(9)
and, therefore, the general solution of equation (9) is:
( , ) ( ) ( )x x
I Rx e e . (10)
Moreover, the Fourier transform of the normal force at the cross-section x is:
2
2
( , ) ( ) ( )x x
I R
A
F x e e
, (11)
where, ( )I and ( )R are the Fourier transforms of the longitudinal strains at 0x ,
corresponding to the incident wave and its first reflection at the free end of the bar, respec-
tively. The propagation coefficient ( ) , the phase velocity ( )C , the wave number ( )k
and the attenuation coefficient ( )a are related as:
( ) ( ) ( ) ( )
( )
a i k a i
C
. (12)
Since one end of the bar is free, the normal force at that end is zero and, consequently, equa-
tion (11) becomes:
( ) ( )( ) ( ) 0d d
I Re e , (13)
where, d is the distance between the strain gauges and the free end of the bar.
3. Experimental.
The determination of the phase velocity and the attenuation coefficient was carried out
using the strain records from a series of impact tests on a PMMA bar impacted by two ball
bearing projectiles of different diameters. The apparatus which was used for the experiments
is shown in Fig. 2. The dimensions of the pressure bar and the projectiles, as well as the
material properties, are given in Table 1.
Fig. 2. Experimental set-up used for the determination of the phase velocity
and attenuation coefficient.
138
Table 1. Projectile and pressure bar dimensions and properties
Projectile / Pressure bar types Material Diameter (mm)
Length
(mm)
Young’s
modulus (GPa)
Tensile strength
(MPa)
Impact
strength
(kJ/m2)
Ball bearing Silver steel 12 - 211 700 1000
Ball bearing Silver steel 20 - 211 700 1000
Pressure bar PMMA 20 1000 2-3 55-80 15
As shown in Fig. 2, four strings are holding the bar in such a way so it can swing freely
along its axial direction. Three pairs of strain gauges were cemented on the bar at different
locations; see Fig. 3, a, in order to examine the change in the shape of the pulse due the ma-
terial damping. Note, also, that each pair of strain gauges formed the arms of a Wheatstone
bridge circuit, as shown in Fig. 3, b.
Fig. 3, a. Location of strain gauges and b Wheatstone bridge arrangements
of the PMMA bar.
There are two active arms in the Wheatstone bridge circuit, the remaining arms are con-
nected to two dummy resistors R1 and R2. GA and GB are the strain gauges, fixed diametrically
opposite on the pressure bar, and Vin and Vout are the bridge input and output voltages. The
strain gauges are sensitive only to the longitudinal strain components, while the bending ef-
fects are minimised. In this case, the measured strain is given by the following relationship:
out
in
4
R
V
S G V A
, (14)
139
where, RA is the number of active arms, S the gauge factor and G the gain setting of the
amplifier. The signals were amplified using three amplifiers type 369-TA, manufactured by
FYLDE Electronic Laboratories Limited. The data recorded from the strain gauges were
captured by a data logger and the pulses were monitored and processed using a PC unit.
4. Results and discussion.
The output of the first impact test using the 12 mm diameter projectile is the voltage
versus time plots shown in Figs 4, a – c.
a
b
c
Fig. 4. Variation of voltage with time from the a first, b second
and c third pair of strain gauges Gi on the pressure bar.
140
As shown, the stress waves generated by the impact were of triangular shape. It is obvi-
ous by observing the graphs that the level of noise increases with the distance between the
point of measurement and the impacted end. Moreover, the waves, that propagate in the two
directions, positive and negative, overlap at the points where the first two pairs of strain
gauges are attached. Therefore, the data recorded from the third pair of strain gauges will be
used in the calculations since the waves do not overlap at that point. Similar results were
obtained from the impact test using the larger projectile of 20 mm diameter.
In order to perform the calculations for the phase velocity and the attenuation coeffi-
cient, the incident and reflected waves, ( )I t and ( )R t must be treated separately as shown
in Figs 5, a, b.
Fig. 5, a incident and b reflected waves.
Using equation (13) it is possible to define a transfer function ( )H , which is neces-
sary for the calculation of the propagation coefficient, as follows:
2 ( )( )
( )
( )
dR
I
H e
, (15)
where, ( )I is the Fourier transform of the axial strain associated with the incident pulse,
which is negative (compressive wave) and, therefore equation (15) becomes:
141
( )
ln 2 ( )
( )
R
I
d
(16)
and, solving for ( ) :
( )1
( ) ln
2 ( )
R
Id
. (17)
The attenuation coefficient can be obtained from the real part of the complex number,
( ) and the wave number from the phase of the transfer function, ( )H after performing
a numerical procedure known as unwrapping [6]. This way, the phase of the transfer func-
tion is determined in the range between 0 and 2. Therefore, before calculating the wave
number, the radian phase angle is corrected by adding multiples of 2, when absolute
jumps between consecutive elements are greater than radians. The wave number is then
expressed as a monotonic continuous function of frequency. The phase velocity can be de-
rived from equation (12) as:
Frequency
( )
Wave number
C
( )k
. (18)
Fig. 6. Experimental phase velocity versus frequency from the impact
generated by the a 12 mm and b 20 mm diameter projectiles.
142
The calculation of ( ) , ( )C and ( )a was carried out using the Matlab code. After
each impact test the voltage records were stored in an ASCII file. The data were sampled at
105 Hz and the number of points used to define the incident wave and its reflections was
29800. The voltage measurements were converted to strain using equation (14). Two double
arrays were created, containing the incident and reflected waves, I and R . The size of
these arrays, which was initially, 1 501 was increased to1 5511 , by adding a series of
zeros to the strain records. This way, the phase velocity and the attenuation coefficient ver-
sus frequency curves were perfectly smooth. The values of ( )I and ( )R were then
determined by taking the fast Fourier transforms of the axial strains, I and R , at the same
time interval, from zero to 5511 microseconds. The next step was to define the transfer func-
tion ( )H , which was obtained by diving the fast Fourier transforms of the reflected and
the incident waves respectively, see equation (15).
The wave number ( )k was obtained from the phase angle of the transfer function after
applying the unwrapping technique. The frequency was then defined in such a way as to
include all 5511 points and the phase velocity was calculated using equation (18). Finally,
the propagation coefficient, which is a complex number, was determined from the natural
logarithm of the transfer function, see equation (17), and the attenuation coefficient was
derived from the real part of ( ) . The attenuation coefficient and phase velocity versus
frequency curves obtained from the impact tests are presented in Figs 6 and 7.
Fig. 7. Experimental attenuation coefficient versus frequency
for a 12 mm and b 20 mm diameter projectiles.
143
The selected frequency range is 0-10 KHz, since in a typical SHPB test does not ex-
ceed 10 KHz. It is also important to note, that the attenuation coefficient and the phase ve-
locity versus frequency curves for both projectiles are very close for frequencies between 0
and 5 KHz. Beyond 5 kHz the scattering increases, since the signal spectrum components
become lower.
Note, also, that the fluctuations observed at higher frequencies are greater for the larger
diameter projectile in which case the pulse generated by the impact is longer, compare
Figs 6, a, b and 7 a, b, indicating that the accuracy of method is increased with the use of
smaller projectiles. The difference in the calculated values of the phase velocity using the
two projectiles for frequencies higher than 5 kHz is at maximum 2% whereas, the effect of
the projectile diameter in the determination of the attenuation coefficient is much more sig-
nificant. It may be observed in Fig. 7, b, that the attenuation coefficient curve which corre-
sponds to the 20 mm diameter projectile drops below zero at a frequency of approximately
9,5 kHz. On the other hand, the result obtained by the strike of the 12 mm diameter projec-
tile is very different as shown in Fig. 7, a, where a much smoother curve is demonstrated.
This implies that in order to study with sufficient accuracy the attenuation coefficient
and phase velocity of the stress waves for the complete frequency range of a SHPB test,
incorporating PMMA pressure bars, the threshold of steel projectile diameters should not
exceed 12 mm. On the other hand, for projectile diameters larger than 12 mm and up to 20
mm the useful frequency range is 0 – 5 kHz.
5. Conclusions.
An experimental technique is presented for the calculation of the phase velocity and at-
tenuation coefficient in a viscoelastic Hopkinson pressure bar by taking into account the
wave dispersion effects. The longitudinal strains of the waves produced by the impact of
ball bearing projectiles on the PMMA bar are measured using three pairs of strain gauges
cemented at different locations on the bar. This way, the propagation coefficient, phase ve-
locity and attenuation coefficient are determined from the Fast Fourier Transforms of the
strains associated with the incident and reflected waves. Thus, C , and are expressed
as monotonic functions of the frequency .
Summarising the main features of the results reported, it may be concluded that the ac-
curacy of this technique strongly depends on the duration of the pulse which is associated
with the diameter of the projectile, and the frequency range of the measurements. Smaller
diameter projectiles generally generate shorter pulses, increasing in this way the accuracy of
the measuring technique. The useful frequency range reported, for sufficiently accurate re-
sults, for the larger diameter projectiles is 0 – 5 kHz.
Acknowledgment.
Acknowledgments are due to Dr J.J.Harrigan of the University of Aberdeen, UK, for his
kind help with the experimental measurements, which were conducted at the University of
Manchester, UK, and his overall supervision for the duration of the project.
Notation:
A – Cross-sectional area of the bar;
RA – Number of active arms;
a – Bar radius;
( )a – Attenuation coefficient;
C – Stress wave velocity in the bars;
( )C – Phase velocity;
d – Distance between the strain gauges and the
free end of the bar;
E – Young’s modulus;
( )E – Complex Young’s modulus;
( )inF t – Force at the end of the input bar;
outF – Force at the end of the output bar;
G – Gain setting of the amplifier;
( )k – Wave number;
L – Bar length;
S – Gauge factor;
t – Time;
inV – Particle velocity at the end of the input bar;
outV – Particle velocity at the end of the output bar;
( ) – Propagation coefficient;
– Complex change in the phase function of the
frequency;
( ), ( ), ( )I R Tt t t – Axial strains associated with the
incident, reflected and transmitted waves respec-
tively;
( , )x – Fourier transform of the axial strain;
– Density;
( , )x – Fourier transform of the axial Stress;
– Angular frequency.
144
Р Е ЗЮМ Е . Експериментально обчислено коефіцієнт поширення, який має зв'язок з дисперсі-
єю хвилі та коефіцієнтом затухання хвилі. Вказані дисперсія та коефіцієнт затухання отримані для
матеріалу ПММА в залежності від частоти в серії дослідів з удару за допомогою стержня Гопкінсона.
де використані кулі у вигляді сферичних підшипників різного діаметру. Застосовано метод корекції
дисперсії хвиль із застосуванням швидкого перетворення Фур‘є щодо поздовжніх деформацій, який
зв’язаний з падаючою та відбитою хвилями.
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Eng. – 1998. – 21. – P. 827 – 836.
4. Zhao H., Gary G., Klepaczko J.R. On the use of a viscoelastic Split Hopkinson Pressure Bar // Int. J. Im-
pact Eng. – 1997. – 19. – P. 319 – 330.
5. Lifshitz J.M., Leber H. Data processing in the split Hopkinson pressure bar tests // Int. J. Impact Eng. –
1994. – 15. – P. 723 – 733.
6. Zhao H., Gary G. A three dimensional analytical solution of the longitudinal wave propagation in an infi-
nite viscoelastic cylindrical bar. Application to experimental techniques // J. Phys. Solids. – 1995. – 43.
– P. 1335 – 1348.
7. Bacon C. An experimental method for considering dispersion and attenuation in a viscoelastic Hopkinson
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From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 05.02.2014 Утверждена в печать 19.02. 2015.
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