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Analysis of a proposed six inch diameter Split Hopkinson Pressure Bar

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Analysis of a proposed six inch diameter Split Hopkinson Pressure Bar
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Jerome, Elisabetta Lidia
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ix, 121 leaves : ill. ; 29 cm.

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Aluminum ( jstor )
Amplitude ( jstor )
Diameters ( jstor )
Inertia ( jstor )
Specimens ( jstor )
Strain gauges ( jstor )
Strain rate ( jstor )
Stress waves ( jstor )
Wave propagation ( jstor )
Wavelengths ( jstor )
Aerospace Engineering, Mechanics and Engineering Science thesis Ph. D
Concrete -- testing ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF
Materials -- Dynamic testing ( lcsh )
Materials -- Fatigue testing ( lcsh )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1991.
Bibliography:
Includes bibliographical references (leaves 82-85)
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Elisabetta Lidia Jerome.

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ANALYSIS OF A PROPOSED SIX INCH DIAMETER
SPLIT HOPKINSON PRESSURE BAR











By


ELISABETTA LIDIA JEROME


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENT FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA


1991












ACKNOWLEDGMENTS


I wish to thank my committee for their support and advice during this
educational process. In particular I thank Dr. C. Allen Ross for his advice,
support, and encouragement. Without him this effort would not have been
possible. I would also like to thank my husband David for his patience and
understanding during the many nights and weekends I spent at my desk or in
front of the computer. I would like to acknowledge the financial support of
the Air Force Engineering and Services Center at Tyndall Air Force Base.
Finally, I wish to acknowledge the help and support of Sverdrup Technology
Inc. during my educational endeavors.













TABLE OF CONTENTS

page

ACKNOWLEDGMENTS .................................................................................... ii

LIST O F FIG U R ES................................................................................................ v

A B ST R A C T .............................................................................................................. viii

CHAPTERS

1 INTRODUCTION.................................................................................... 1
O bjective.............................................................................................. 1
Background .......................................... ................ .............................. 1
A approach .............................................................................................. 3

2 MATHEMATICAL BACKGROUND ...................................... .......... 5
SHPB Assumptions ........................................................................... 5
Equations of Motion ............................................. ........................ 8
Solutions of the Frequency Equation............... .......... ........... .. 11
Transient Behavior............................................................................ 19

3 MATHEMATICAL AND NUMERICAL ANALYSIS............................ 21
Scaling Law s.......................................................... ............................. 21
Wave Propagation in an SHPB .......................................... ........... .. 25
Modified Pochhammer-Chree Method....................................... 26
Finite Difference Numerical Simulation..................................... 35
Experimental Method ............................................................. 37
Com prison of R results ........................................................................... 38
Inertia and Friction Effects ............................................................. 70

4 SIX-INCH SHPB FEASIBILITY .............................................................. 78
Sum m ary ............................................................................................... 78
Conclusions and Recommendations ................................... ........... 80









page

-.............................................. 82


REFERENCES

APPENDICES


A Modified Pochhammer-Chree Program and Sample Outputs......... 86
B Ratio of Fourier Coefficients for Given a/A of 3 Bar Diameters...... 105
C Miscellaneous HULL Plots ................................................................... 107
D HULL Sample Input Files ..................................................................... 114
E Fortran Listing for Inertia Correction............................................... 117

BIOGRAPHICAL SKETCH ....................................................................................... 121













LIST OF FIGURES


page

1. Compressive SHPB Schematic ................................................... ............... 1
2. Cylindrical Coordinates ..................................................... ......................... 9
3. Pochhammer-Chree Solutions for First Three Modes................................... 15
4. Tyndall 2-inch SHPB. Experimental Trace. Aluminum Specimen
Length and Diameter 1.254 inch ...................................................... ............. 30
5. Modified Pochhammer-Chree Results. Incident Stress Normalized
to 1.0.................... ........................................................................................ .. 3 1
6. Normalized Longitudinal Stress Given as a Function of Radial Location for
Several Fourier T erm s .......................................................................................... 34
7. Experimental Results. 2-inch SHPB. Stress Pulses for a 6061-T6511
Aluminum Specimen, Diameter = 3.18 cm, Length = 3.18 cm.................. 39
8. Experimental Results. 2-inch SHPB. Stress Pulses for a 6061-T6511
Aluminum Specimen, Diameter = 2.54 cm, Length = 2.54 cm.................. 40
9. Experimental Results. 2-inch SHPB. Stress Pulses for a 6061-T6511
Aluminum Specimen, Diameter = 0.56 cm, Length = 0.56 cm.................. 41
10. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY) Pulses
for an Aluminum Specimen, Diameter = 3.18 cm, Length = 3.18 cm....... 42
11. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY) Pulses
for an Aluminum Specimen, Diameter = 2.54 cm, Length = 2.54 cm........ 43
12. HULL Output. 2-inch SHPB. Stress (TYY) and (EYY) Strain Pulses
for an Aluminum Specimen, Diameter = 0.56 cm, Length = 0.56 cm........ 44
13. Experimental Traces at Three Incident Stress Levels (60 MPa-Top,
100 MPa-Middle, 167 MPa-Bottom). Specimen Diameter
1.253 in., Length = .625 in., 2-inch SHPB.................................. ............ 46
14. HULL Output. 2-inch SHPB. 2-inch Diameter and Length,
Aluminum Specimen, 257 usec Pulse ......................................................... 48
a 1
15. Modified Pochhammer-Chree Results. A 125 6-inch SHPB.
A 125 6-inch SHPB.
750 sec Pulse.................................................................................................. 49
a 3
16. Modified Pochhammer-Chree Results. A 250 6-inch SHPB.
500 pusec Pu lse ..................................................................................................... 50







pWag


a 3
17. HULL Output. a = 250 500/sec Pulse, Station 1
(Top) Bar Center, Station 2 (Bottom) Bar Surface ....................................... 51
a 3
18. Modified Pochhammer-Chree Results. A-125 6-inch SHPB.
250 usec Pulse.. .............................................................................................. 52
a 3
19. HULL Output at Incident Bar Strain Gage. a = 125' 6-inch
SH PB 250 ysec Pulse.......................................................................................... 53
20. Experimental Traces at Three Incident Bar Locations for Two Striker Bar
Lengths. 16-inch (Top) and 25-inch (Bottom), 2-inch SHPB .................... 55
21. HULL Output. 2-inch Bar, 250 /sec Pulse, All Stress Components
on the Surface at Incident Bar Strain Gage.................................. .......... .. 56
22. HULL Output. 6-inch Bar, 500 usec Pulse, All Stress Components
on the Surface at Incident Bar Strain Gage.................................. .......... .. 57
23. Modified Pochhammer-Chree Results at Incident Bar Strain
G age. 2-inch Bar ............................................................................................ 58
24. HULL Output. 6-inch Bar, 250 usec Pulse.
Surface (Top) and Center (Bottom) Transmitted Stresses ........................... 59
25. HULL Output. 6-inch Bar, 500 ,usec Pulse.
Surface (Top) and Center (Bottom) Transmitted Stresses ........................... 60
26. HULL Output. 2-inch Bar, Longitudinal Stress Near the
Impacted End (Top) and at the Strain Gage Location (Bottom) ................. 62
27. HULL Output. 6-inch Bar, All Stresses Near Impacted End (Top)
and at the Strain Gage Location (Bottom) .................................. ............. 63
28. HULL Output. 2-inch Bar, All Stresses Near the Impacted End (Top)
and at the Strain Gage Location (Bottom) .................................. ............. 64
29. HULL Output. 2-inch Bar, All Stresses at Three Locations
at the Bar-Specimen Interface ........................................................................ 66
30. HULL Output. 2-inch Bar, Stresses in Aluminum Specimen
at Two Locations Near the Surface (Top), Near the Center (Bottom).
Compare with Figure 29..................................................... .................................... 67
31. HULL Output. 2-inch Bar, Transmitted Stresses at Three Radial
Locations. Surface (Top), Middle (Middle), Center (Bottom).
Compare with Figure 29 ..................................... .......................... .......... 68
32. HULL Output. 2-inch Bar, Transmitted Stress with Zero Friction
Between Specimens and SHPB ............................................. .............. 74
33. HULL Output. 2-inch Bar, Transmitted Stress with Infinite Friction
Between Specim en and SHPB......................................................................... 75
34. HULL Output. 6-inch Bar, Transmitted Stress with Zero Friction............ 76









page


35. HULL Output. 6-inch Bar, Transmitted Stress with Infinite Friction....... 77
36. Modified Pochhammer-Chree Results. Radial Variations of Reflected
Stresses. 2-inch SHPB (Top), 6-inch SHPB (Bottom) ................................ 88
37. Modified Pochhammer-Chree Results. Radial Variations of Transmitted
Stresses. 2-inch SHPB (Top), 6-inch SHPB (Bottom).................................. 89
38. HULL Output. 2-inch SHPB. Radial and Longitudinal Velocities
at Incident Bar Specimen Interface. (Bar Surface: Top,
Middle of the Bar: Bottom, Bar Center: Right)............................................ 108
39. HULL Output. 6-inch SHPB. All Transmitted Stresses at Bar
Surface.............................................................................................................. 109
40. HULL Output. 6-inch SHPB. All Transmitted Stresses at Bar
Center. Compare with Figure 37 ...................................................................... 110
41. HULL Output. 2-inch SHPB. 1-inch Aluminum Specimen.
Incident and Reflected Stresses............ ................................................. 111
42. HULL Output. 2-inch SHPB. 1-inch Aluminum Specimen.
All Stresses W within the Specimen ..................................................................... 112
43. HULL Output. 2-inch SHPB. 1-inch Aluminum Specimen.
All stresses in the Transmitter Bar................................................................. 113







Abstract of Dissertation Presented to the Graduate School of the
University of Florida in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy


ANALYSIS OF A PROPOSED SIX INCH DIAMETER
SPLIT HOPKINSON PRESSURE BAR

By


Elisabetta Lidia Jerome


August 1991

Chairman: C. Allen Ross
Major Department: Aerospace Engineering, Mechanics and Engineering
Science


The objective of this investigation was to determine whether a 6-inch
diameter Split Hopkinson Pressure Bar (SHPB) is feasible; whether the
assumptions made in a typical 2- to 3-inch SHPB still apply; and what chan-
ges, problems, and issues would be associated with an SHPB that size. The
effort included a mathematical and a numerical analysis. Experimental data
from a 2-inch SHPB were used to compare and validate analytical models.
A methodology based on the Pochhammer-Chree frequency equation
was developed by the author to study the wave propagation and wave inter-
action in any size SHPB. Additionally, a second order accurate finite dif-
ference code was used to mathematically understand the wave motion in a
typical bar.
The effects of friction and inertia were analyzed numerically, and
general guidelines and corrections were developed.
This investigation concluded that a 6-inch SHPB is theoretically
feasible and, in fact, identical to any existing system if the input pressure
pulse duration is scaled appropriately.







The modified Pochhammer-Chree method can be used to assess the
validity of any size SHPB and to study the wave interactions. If the specimen
length and diameter are kept in the same order of magnitude, one need not
worry about friction in the proposed 6-inch SHPB. A correction will be
needed to account for inertia, especially at strain-rates of 50 to 100 sec' and
above. The Split Hopkinson Pressure Bar has been and remains a reliable,
accurate, and useful tool to study material response to loading, even for a
new proposed system six inches in diameter.













CHAPTER 1


INTRODUCTION


Objective


The objective of this investigation is to determine whether a 6-inch
diameter Split Hopkinson Pressure Bar (SHPB) is feasible; whether the
assumptions made in a typical 2- to 3-inch SHPB still apply; and what major
changes, problems and issues would be associated with an SHPB that size.


STRAIN GAGE


INCIDENT BAR


STRAIN GAGE


TRANSMITTER BAR


Figure 1. Compressive SHPB Schematic




Background


The Split Hopkinson Pressure Bar may be used to study materials at
high strain-rates in tension, shear and compression. A compressive SHPB is
shown schematically in Figure 1. An SHPB system consists of a specimen
sandwiched between and in contact with two elastic bars--incident and trans-
mitter bar. The incident bar is impacted by a striker bar at a known velocity


V






STRIKER BAR







causing a pressure wave to travel down the bar. This wave is partially trans-
mitted and partially reflected at the incident bar/specimen interface, and
partially transmitted and partially reflected at the specimen/transmitter bar
interface. Strain gages attached to the incident and transmitter bars record
the strains associated with the reflected, transmitted, and incident pulses.
From these strain data, stresses, strains, and strain-rates in the specimen can
be computed as a function of time.
Although different sizes of SHPBs have been built, the largest one in
existence in the USA is 3 inches in diameter. The main reason behind this
size limitation is that it is generally assumed that the wave propagation in
the bar is one-dimensional, implying a high length to diameter ratio. A
simpler form of the SHPB has been around since the late 1940s, when
techniques and instrumentation were not as sophisticated as they are today.
Historically, there may not have been a need to test a material specimen
larger than 3 inches in diameter. However, in recent times the need for
more economical structures to resist impulsive loadings associated with
earthquakes, accidental explosions, and effects of conventional explosives
has prompted more research into strain rate effects on the properties of
concrete. Considerable data have been generated in 2- to 3-inch diameter
SHPB systems on concrete specimens containing small or scaled down
aggregate. The question which arises from such small tests is, does the
strain rate sensitivity, found for the small laboratory specimens, apply to the
standard 6-inch diameter large aggregate test specimens which are cored or
cast in the field? There is a need to use a 6-inch diameter SHPB type
apparatus to test concrete specimens, which will match the concrete
specimens which are cored directly in the field.
The SHPB has proven to be a very effective and useful tool to conduct
high strain-rate material response studies since Kolsky [1] described its use
in 1949. Detailed discussions of how a typical system works can be found in
numerous papers including Hauser, Simmons and Dorn [2], Lindholm [3],
Malvern and Ross [4], Robertson, Chou, and Rainey [5], and many others.







Because of the inherent difficulties associated with a system of this
kind, an idealized one-dimensional wave theory is commonly assumed where
stress is uniform over each cross section, plane cross sections remain plane
during motion, and stress and strain are uniform throughout the length of
the specimen. The above assumptions are adequate if the length of the bar
greatly exceeds the diameter, in order to ensure no end effects and the
loading pulse is much larger than the transit time of the specimen. However,
factors like dispersion, radial and axial inertia effects, and friction between
the specimen and the incident and transmitter bars can cause erroneous
results if care is not used in either conducting the experiment or analyzing
the data. Approximate corrections for the calculated one-dimensional stress
have been formulated to account for inertial and frictional effects. Kolsky
[6] introduced a correction for radial inertia. Davies and Hunter [7] dis-
cussed both friction and inertia effects. Other papers of interest on this
subject are those by Rand [8], Dharan and Hauser [9], Samanta [10],
Jahsman [11], Chiu and Neubert [12], and Young and Powell [13].
The original SHPB apparatus was devised to study materials in com-
pression. More recently it was modified to test specimens in both tension
and torsion. Examples of these techniques are discussed by Nicholas [14],
Lawson [15], Baker and Yew [16], Jones [17], Okawa [18], Ross, Nash, and
Friesenhahn [19], Rajendran and Bless [20], and Ross [21]. Follansbee and
Frantz [22] used the Pochhammer-Chree frequency equation to correct the
recorded pulses for dispersion as they travel down the bar assuming only the
first longitudinal vibrational mode is excited.


Approach


Before undertaking any research it is imperative to know and under-
stand what has already been done on the subject. This ensures no duplica-
tion of work and gives a good working foundation from which to start the
research effort.








The wave propagation in an SHPB is governed by the equations of
motion. Generally, it is assumed the motion is one-dimensional, and there-
fore is uniform in the radial and the circumferential direction. Although the
one-dimensional wave theory has been shown to be accurate for the existing
systems, a mathematical analysis is needed for the proposed 6-inch SHPB to
assess whether this simple theory still applies. The two-dimensional axisym-
metric cylindrical equations of motion need to be analyzed and solved
(approximately), by various methods to understand the wave motion in a bar.
There are no known exact solutions to the two-dimensional wave
propagation problem in a finite cylindrical bar, and therefore one must turn
to a numerical scheme, namely a finite difference approximation. Modern
computers make it possible to solve large problems in a reasonable amount
of time. The plan was to first numerically simulate an existing SHPB in
order to compare the results with experimental data and have confidence in
the reliability of the numerical approximation. The second step would then
be to simulate the proposed 6-inch system and gather information on the
wave motion.
The final step will be a comprehensive look at the different
methodologies, approximations, and assumptions to better understand the
implications of increasing the size of the SHPB system. From this one can
assess whether the data reduction of the current SHPBs still applies to the
large bars, and what are the problems associated with a system two or three
times the size of the existing ones.













CHAPTER 2

MATHEMATICAL BACKGROUND


SHPB Assumptions


As discussed earlier, a one-dimensional wave theory is generally
assumed in the analysis of an SHPB. The one-dimensional theory assumes
longitudinal spatial variations only. Applying Newton's law for a displace-
ment u in the x direction of the diagram below, gives the equations of motion
for an elastic system


a ---


8a
---- +- dx
ax


I-- -- dx


aa (a2u
-aA + (a + dx)A = pAdx
ax atz

or upon simplification
au a2u
ax =p at2


and Hooke's law


du
= Ee =-E,
ax







where a is the stress, E is Young's modulus, E is strain, t is time, A is the
cross-sectional area, and p is the density of the material. Equation (la) can
be rewritten as follows

a2u / 2U
Wt (E/p ) ax2 (lb)


where E/p is the familiar elastic wave speed squared, or E/p =Co The
general solution to (Ib) is of the form


u = f(x + Cot) + g(x Cot)


corresponding to forward and backward propagating waves. Assuming
u = g(x Cot) only, differentiate u and obtain


Ou u ,
Sg' and -cog
ax at


where g' denotes differentiation of the function w.r.t. the argument (x-Cot).
Eliminating g' between these two expressions yields



Bu 9u
S= --Co = -CoE
at ax


Strains are usually measured in an SHPB experiment at some distance from
the specimen. To obtain the interface displacements, these recorded strains
must be time-shifted to give the strains at the interfaces, and then integrated
over time to give

S= f Co (ei r) dt and u2 = Co tdt







where the subscripts i, r, and t indicate incident, reflected and transmitted.
Additionally, ul and u2 are the specimen's displacements at the incident and
output bars interfaces respectively. The negative signs give positive dis-
placements for compressive (negative) strains. Let L be the length of the
specimen; then the average strain in the specimen is


Es = (U2 Ui)/L.


This is sometimes approximated for short specimens and slowly changing
stress by assuming equal stresses at the two interfaces, which implies that


Et = Ei + Er.

The specimen strain then becomes


s= (-2co/L) f trdt


and the specimen strain rate is

s = (-2co/L) Er.

By using Hooke's law, the stresses on either face of the specimen are


os2 = E A and Usl = E (Ei + Er)


A
where is the ratio of the bar and specimen cross section areas. If the
As
interface stresses are not assumed to be equal, then Et # Ei + Er and the
specimen strain and strain rate are given by



S=( Co/L) f ( i +Er) dt and s = (-co/L) (E i + r)







There are some problems associated with such a simplified one-
dimensional theory, even for long slender rods, and these problems are
accentuated for larger diameter and shorter length SHPB systems. First,
longitudinal waves are dispersive, (wave speed is a function of frequency),
which means the waveform that is recorded at a distance from the specimen
is not the same waveform that actually arrives at the specimen. Wave disper-
sion can be corrected for, but is time consuming and the correction methods
are not exact. The one-dimensional theory also neglects stress and displace-
ment variations across the cross section. Since strains are measured only at
the surface, this can also lead to errors.


Equations of Motion


To assess the validity of this simple theory, one can study the equa-
tions of motion that govern the wave motion in a circular cross section
SHPB. The equations of motion are given in tensor form as



d2u
V-T+pb= dt2 (2)



where T is the stress tensor, b are the body forces (which are usually safely
d2
neglected), and y is the acceleration term. In cylindrical coordinates

(Figure 2), equation (2) becomes


02Ur dUrr rr UOO 1 UrO 8Urz
p + +
at2 8r r r 80 Oz


p + r+ +2 (3)
at2 8r r aO 8z r


Suz a"rz + 1 BOoz + a0z +g
at2 Or r -0 aZ r
























Figure 2. Cylindrical Coordinates


Equations (3) represent the three-dimensional motion in a cylindrical
bar. The modes of vibration associated with an isotropic circular cylinder
are in general very difficult to calculate and they depend on the specific
boundary conditions as well as on the form of the input loading. If the axis
of the cylinder coincides with the z-axis, two general types of vibrational
modes exist. One set is symmetric and one is asymmetric. In this analysis,
the assumption is made that since the striker bar impacts the incident bar
squarely, then the motion should be axisymmetric. This means the solution
is independent of theta; then u., a, and oz vanish, and the second of equa-
tions (3) is satisfied identically. It also means that the only possible modes
of vibration will be symmetric. Using Redwood's [23] notation, these
modes will be called Mm,n where m is either 1 (symmetric) or 2 (asymmetric)
and n ranges from 1 (Young's modulus mode) to infinity. Therefore, in this
study of an SHPB, only M1,n modes are considered. The constitutive rela-
tions for an elastic material can be written as


aur (Ur az
St +2 U)r (fr aUz )
Or =('t+ ")fi 7+-f







9Ur Uz)
arz a= ut + u)
az ar
(4)
ouz (r OUr
uz = (1 + 2u)z + +r


aOrr aUrz
aeeoo = r- + + rr
Or 9z

where A and P are Lame's constants. Making use of (4) and neglecting theta
dependence, equations (3) can thus be reduced to only two equations


2Ur [2ur 1 Our Ur 2Uz a2ur
p = ( + 2) + (2 2+u) --+
at2 ar r Or r rz az2

(5)
2z 2Uz 1 Uz O2Uz 2Ur 1 OUr
p- =2 2 r + + 2,u) + ( + u) ++r r -
t r r Or 0z2 L-z r 9z (5


The boundary conditions require no stress at the bar surface or

arr =0 and Or = 0 at r = a


where a is the radius of the bar. Also at a free end, Urz = 0 and Ozz = 0.
For a colinear impact of one bar against another, the initial conditions
can be in the form of a step change in pressure


az = -PH(t), arz = 0


where P is a uniform pressure and H(t) is the Heaviside step function, or as
a harmonic displacement of uz = sinw t and arz = 0. The condition orz = 0 is
correct only if the two bars are of identical diameter and material, so that
the interface is a plane of symmetry of the stress state, until a reflected wave







reaches the interface. Equations (5) have been solved for an infinitely long
bar by Pochhammer and Chree for the propagation of sinusoidal waves and
studied by numerous investigators over the years [6, 24, 25, 26, 27, 28, 29].
The solution is called the frequency equation and has the form



2y Jo(ha) (2y2 P) Ji(Ka)
a = 0 (6)
2u 2 Jo(ha) Jo(ha) uy-a JI(Ka)
aa A + 2/ aa




where h2 = (2r/A)2 [pc /(A + 2u) 1],K2 = (27r/A)2 [(pc/p) 1], A = wave-

length, y = 2 Jo and J1 are the zero and first order Bessel functions
A Cp
respectively, w is the angular frequency, and cp is the phase velocity.


Solutions of the Frequency Equation


There are an infinite number of solutions to Equation (6), each cor-
responding to a unique mode of vibration. The solutions are exact only for
an infinitely long cylinder although, for a cylinder whose length greatly
exceeds its diameter, the errors are small. Furthermore, the solutions are
not explicit but rather a function relating the phase velocity cp, the
wavelength A and Poisson's ratio v. Now (6) can be written in implicit
form as

2Uy -J1 (Ka) 2y Jo (ha) 2y2 J (a).

(7)
2u 2Jo(ha) 2 Jo (ha) =0
[ a2 A + 2/s







By making use of the following properties of Bessel functions,

a (Jo(ha))= -Jl(ha)


A (Ji (ha)) = hJo (ha) (J (ha)) /a,


equation (7) can be rewritten as


4py (KJo(Ka) Ji(Ka)/a) (-KJi(ha)) ((2y2 ) Ji (Ka)


L2p. (-hJo (ha) + J1 (ha)/a) +2Jo (ha) = 0,
A + 2,u

which finally becomes


y2 JK o(a) 1 1 2+) + (K- 2) Jo(ha)= (8)
J1 (ca) 2 a 2 hJ1 (ha)

Equation (8) has been solved numerically by several investigators
[25, 27, 30]. It involves assuming a value of Poisson's ratio and then for
varying values of frequencies, ranging from zero to some large number,
equation (8) is numerically evaluated to obtain the values of the phase
velocity. This turns out to be a very laborious task even with the aid of high
speed computers. Results are usually shown in graphical form, namely in
curves relating the phase velocity and the frequency, both in nondimensional
form for general applicability. Figure 3 [22] shows the solution to equation
(8) for a Poisson's ratio of .29. Only the first three vibrational modes are
plotted namely M1,1, M ,2 and M ,. Recall that equation (8) only applies to
symmetric vibrational modes. Mode M is of most interest since it appears
to be the only mode excited in a typical SHPB. As the frequency of this
mode approaches zero, or as the wavelength approaches infinity, the phase
velocity approaches (v- ), which is the elastic, one-dimensional bar-wave







speed, co. As the frequency becomes very large, the phase velocity ap-

proaches cr, the velocity of Rayleigh surface waves. Although it may not be
clear from Figure 3, Cp, the phase velocity, approaches cr from the down side,
implying that the phase velocity reaches a minimum which is less than the
Rayleigh wave speed at some intermediate frequency. Recall that


2 A+2 2 2 E
Cd- Ct- Co -
P P c P

h2 = (27t/A)2 (pc/(A + 2,u) 1) and K2 = (23r/A)2 (pc/ 1)


and it is easily shown that co < ca and for M,1 cp < co. Then for the M1,1
mode the parameter h is always imaginary since, for all frequencies, cp < ed
where Cd is the dilatational wave speed in an infinite medium.
a
For very low frequencies K is real, but past roughly = 0.65, when c

becomes less than the shear wave speed in an infinite medium ct, K is imagi-

nary. This means that at low frequencies ( < 0.65), the displacement is

mainly due to plane transverse waves, since the set of dilatational waves
exists only as a surface disturbance. At high frequencies the total displace-
ment becomes increasingly like a pure surface disturbance [33].
Mode M1,2 and higher may be of interest in a "larger" SHPB and at
high frequencies. These modes have what are called cut-off frequencies,
i.e., frequencies at which the phase velocity becomes infinite and no real
disturbance propagates in that mode. From equation (8) this implies either
that y = 0.0 or that J1(Ka) = 0.0. If y = 0.0, equation (8) becomes


1 o 1 1 2 Jo (ha)
2 ct a 2 c2 hJ1 (ha)







since


ic-=--- 1 = 1~ 1~ ,
2 2 2
K2 ) C2



and K 2 = -O for Cp = oo, where wco is the cut-off frequency.
ct
2
Following the same reasoning h = -, for cp = oo and thus one can write
Cd"


tJO (ti 0a (9)
\Jo -a
c2 Wco J1 ad
c Cd Cd



If Ji (Ka) = 0.0, it can be said that



J, (Ct a = 0.0 (10)
[Ct )

Equations (9) and (10) can now be used to determine the cut-off frequencies
for any mode M,n .
From the solutions of the frequency equation, equation (8), it is
possible to calculate the radial and longitudinal displacements for each
mode at every frequency, and Poisson's ratio within the cylinder. Bancroft
[25] calculated and plotted the amplitude of the longitudinal displacement
a
and showed that they are (as expected) uniform for =0.0 and the motion

a
is confined to the outside surface when approaches infinity. This agrees

with the observations made earlier about mode M,.1 response in general.

Numerous other approximate methods of varying degrees of com-
plexity have been developed to mathematically treat an SHPB system. They
fall somewhere in between the simple one-dimensional theory and the two-
dimensional, coupled differential equations just presented. Unfortunately,








those that show good agreement with the Pochhammer-Chree solution have
very complicated descriptive equations, while those simple equations agree

with the exact theory only over a very limited frequency range.


1.4



1.2
Cd/Co
o ^ MODE M1
0 1,3 C
0 1.0 P= =

SM Cr = RAYLEIGH VELOCITY

0.8 C2=
MODE M P
C/Co 1,1

0.6



0.4
0.0 0.5 1.0 1.5 2.0
a/A



Figure 3. Pochhammer-Chree Solutions for First Three Modes.


When computers were not readily available and usable, a large num-

ber of approximate theories were developed. Most important to this study

are the different methods and approaches used to derive these theories. As

already discussed, the simplest analysis is to assume purely axial stress,
uniform over each cross section. This leads to the familiar one-dimensional

wave equation in terms of the axial displacement u,,



2Uz a2 (11)
at2 pz







which predicts that waves of all frequencies travel at the same constant
velocity co = v'O. A better approximation to this theory introduces a cor-
rection for the radial motion by considering the inertia of the cross section
[23]. The approach involves the use of Hamilton's principle which states
that the first variation of the integral of the Lagrangian, (T V), with respect
to time is zero. Specifically


6 f (T- V) dt =0 (12)


and (T V) can be expressed as




f0 pA -+(vR) 2 a EA a dz ] (13)
L0 Iii pA ( at ( az atj -^j2 az


kinetic energy potential (strain) energy


where L is the length of the cylinder, R is the radius of gyration about the
z-axis, v is Poisson's ratio, and the velocities in the z and r directions are
respectively


_uz 8ur
au_ and aur
at at


az
But ur can be assumed in this approximation to be of the form -v r-z so

that


8ur a-Uz
S- -vr t
at 0z 0t "







The result of substituting equation (13) into (12) and solving via
integration by parts is as follows


a2Uz 2 a4uz a 2u
P a-- (vR) a2a2t- =E (14)
P at2 t Z2 ( 14




This equation gives a better approximation of the exact theory than
equation (11) at low frequencies. However, for short wavelengths the errors
become considerable.
A third approximate theory developed by Love [29] is based on the
exact characteristic equation (8), where the Bessel functions are expanded in
a power series. If a, the radius of the cylinder, is small enough so that ha and
Ka are small compared to unity, then powers of ha and Ka higher than the
second can be neglected. That is


Jo(Ka) = 1 4 (Ka)2


Ji(Ka) = V2 (a)

and the phase velocity becomes


Cp = 1 v2a2) (15)




and may be rewritten as



v where c a (16)
S=1 v2,2- where Co= V p
Co X)







a
This theory agrees fairly well with the exact theory up to values of = 1 but

then rapidly diverges. The assumptions that Kca and ha are small compared to
unity imply that the wavelengths of the vibrations are large compared to the
radius of the cylinder.
Another common theory was developed by Mindlin and Herrmann
[26]. It considers shear stresses and strains by assuming first that the radial
displacement is of the form


Ur = (r/a) u(z,t)


with us = 0 and uz = w(z,t). Forces and moments are then calculated from
standard engineering mechanics formulae and corrected for shear and iner-
tia by introducing factors K1 and K2. This leads to equations of the form



2 2 2u 2 22 Au 2 a2U
a Kp --- 8K2( + ) u 4aK2 pa 2
az2 z 3t
(17)
au 2 a2a) 2 232)
2aa + a2 (A + 2u) =p a
az az" at
By substituting


u = A exp (-iyz) exp (iwt)


and
w = B exp (-iyz) exp (iwt)


into equation (17), two equations in A and B are obtained, from which A and
B can be eliminated to again obtain a characteristic equation which relates
phase velocity and frequency for the first two modes. By adjusting K2 and
K1, a very good approximation can be obtained for mode M1,i Mode M1,2,
on the other hand, shows considerable deviation from the exact theory.







Many other theories have been developed and, in general, their com-
plexity is directly proportional to their accuracy. These analyses are impor-
tant because of the insight they give into the response of a cylinder subjected
to impulsive loading. Knowing how stresses and displacements are dis-
tributed is crucial to the study of wave propagation and SHPB systems.


Transient Behavior


Thus far, the discussion has been limited to continuous waves or at
least to pulses more than just a few cycles in length. However, in an SHPB
experiment, a very short pulse is propagated through the cylinder, giving rise
to a transient behavior which cannot be completely determined from the
continuous theories; therefore, a different approach must be taken. One
way to tackle the transient problem is to use Fourier analysis. By decompos-
ing the pulse into its continuous component, continuous wave theory can be
used on each component at any point down the bar and the new pulse can be
obtained by adding the pieces together again. This method is not exact, but
it does give a close approximation, especially at some distance from the
source.
Choosing a mathematical function that describes the input pulse and
that can be represented by a reasonably simple Fourier series is very dif-
ficult. Davies [27] used a trapezoidal pulse while Kolsky [28] assumed an
error function. In both cases the Pochhammer-Chree curves are used to
obtain the phase velocity for each pulse, and in calculating the new shape of
the wave-forms, only the fundamental mode M1,1 was assumed to be ex-

cited.
Another approach taken by some investigators in studying transient
behavior is based on the concept of "dominant groups" and the method of
stationary phase. The idea is that if at the beginning everything is in phase,
any time after that all waves are out of phase, and therefore interfere
destructively. When the combined effects of all these waves are studied, the





20


main contribution comes from a small "dominant" group whose phase
velocities, periods, and wavelengths are almost the same. By focusing on
this special group, one can obtain approximate propagation of longitudinal,
flexural, and torsional waves in a cylinder.
In this study a modified Pochhammer-Chree method is developed
using the Fourier analysis approach. It is straightforward and a much
simpler numerical simulation than the existing complicated computer codes.
This new approach is presented in the next section.












CHAPTER 3


MATHEMATICAL AND NUMERICAL ANALYSIS


Scaling Laws


In order to analyze the propagation of waves in a cylindrical bar, one
should ideally solve the three-dimensional equations of motion. Unfor-
tunately, there is no known closed form solution to the problem. Even if
axisymmetry is assumed, the two-dimensional equations of motion cannot be
easily solved. The problem is further compounded when a different material
is introduced, such as when a specimen is introduced in an SHPB. The
addition of two interfaces between dissimilar materials creates reflections
and transmissions which cause waves to superpose, cancel each other, and in
general interact with each other in a very complicated way. Pochhammer
and Chree independently solved the equations for the propagation of a
sinusoidal wave in an infinite bar. The solution, discussed in chapter 2, is
the so called frequency equation; it has an infinite number of solutions, one
for each mode, and results in a function relating phase velocity, wave length
and Poisson's ratio. These solutions are valid and exact for infinitely long
bars. If the bar is long enough to eliminate end effects (approximately 10
diameter lengths) and if the interest is in the transient pulse in its first
passage, then the infinite assumption is reasonable.
The Pochhammer-Chree solutions can be plotted in nondimensional
form as shown in Figure 3. It is interesting to note that since A = CpT, then

a a ac a
S and thus c where T is the pulse duration and is defined
as T /c, where L is the striker bar length. Now if a is multiplied by a
as T = 2 Ls/co, where L is the striker bar length. Now if a is multiplied by a








constant factor and T is also multiplied by that same factor, then a plot of

a vs. -- will not change. What this means is that mathematically, under
0
the assumptions made, if the diameter is multiplied by a constant and the
duration of the input pulse is simultaneously multiplied by the same con-
a a
stant, then the solution for the original a is the same for both values.

This applies to a sinusodial, continuous input where the wavelength is con-
stant, and travels at only one wavespeed relative to the bar radius: a long
wave in a large bar will have the same wave speed as a short wave in a small
a
bar, as long as -A stays constant.

What happens if the input is a transient pulse? A simple mathemati-
cal function can be chosen to describe this type of input which can be
represented by a Fourier series. A transient pulse is composed of a
spectrum of frequencies; the higher frequency components travel more slow-
ly than the lower frequency components, and thus lag behind and cause the
initial sharp pulse to spread. This spreading is called dispersion. The Poch-
hammer-Chree solutions give the velocity of each wave depending on its
frequency. Thus, one may correct a given pulse for dispersion by repre-
senting the pulse by a series of frequency components, calculating how far
each frequency component has traveled in a certain time, and then reas-
semble the pulse. This task basically amounts to correcting for phase chan-
ges within each term of the Fourier series. It is important to be able to
correct for dispersion since the SHPB pulses are recorded on strain gages at
distances typically 30 to 60 inches from the specimen. Interest is in the
response at the specimen itself, but in many cases a strain gage cannot be
physically placed there. The dispersion correction technique allows one to
predict the shape of the pulse as it travels from the specimen to the strain
gage.







Further, following the same kind of reasoning as for the continuous

wave case, a C a t can be written where n is the number of points
A, c nAtco

taken to represent the transient pulse and At is the time interval between

them. In essence, nAt is the period of the transient pulse, and if a
n At
stays constant, the case is analogous to the continuous case. This means as
long as the bar radius and the input pulse length are increased or decreased
by the same amount, the response characteristics of a SHPB system will not
change.
This fact is important to this study because the main concern is the
feasibility of a 6-inch SHPB system. What the Pochhammer-Chree equa-
tions show is that if the incident pulse is of large enough duration relative to
the bar radius, then, as far as the longitudinal waves of the M,, mode are

concerned, there is mathematically no difference between a 2-, 3-, 6- or
60-inch diameter SHPB. One must then determine what pulse duration
would be needed to make a new 6-inch system equivalent to the existing 2-
and 3-inch bars and whether the stress, strain and displacement distribution
of that pulse are satisfactory for a usable SHPB.
As mentioned earlier, for a typical SHPB system, a uniform stress and
strain distribution is assumed, and only strains on the surface are measured.
It is then very important to identify under which conditions the one-dimen-
sional assumption is acceptable. Davies [27] computed the ratio of the
longitudinal displacement at the surface to that of the longitudinal displace-
a
ment along the bar axis and showed that for less than 0.1 their dif-

ference is less than 5 percent. Thus for a continuous pulse, with one value of
a
A, one-dimensionality is assured if A < 0.1. On the other hand, a tran-

sient wave contains many different frequencies, and in fact the majority of

the a s will be greater then 0.1. Thus, if the transient wave f(t)is








represented with a Fourier series expansion, we would generally express it
in a form such as


A o0
Ao + Dncos(nwot-qo'.)
f(t)- 2 Dn cos (n ao t -
2 1


where Ao and Dn are Fourier coefficients and p'n is the phase angle which
is a function of the wave speed and the wavelength. Specifically, from [31]


=c(ncox Co
Pn = (P+ (nwx) [( 1i
o c [-n- J

where nwo is the angular frequency and c, is the phase velocity of the nth
component. Equation (8) and Figure 3 give the relationship between
c, and the frequency. The amplitudes in a Fourier series expansion are in
decreasing order, meaning that Ao is the largest and Dn+1 is less then Dn.
Furthermore, the frequencies of each component are also in order, they start
at zero and go to infinity. In summing the Fourier terms, each term will
contribute less and less to the overall pulse. This is why truncating the series
at some large number, say 55, results in a very small error. Also, at some
a
point the nth component will have an A 0.1 To ensure one-dimen-

a
sionality in a SHPB the terms preceding the one with A = 0.1 must sum up

to give almost the complete pulse. In other words, the amplitude D, of the
a
nth component having =0.1 must be a small fraction of the original

amplitude Ao. Notice that it does not matter at which component
a
X reaches 0.1, but rather the value of the amplitude at that point must be

small. If it is at the 10th or the 100th term in the Fourier series it makes no
difference. The only question then is how small should this Dn amplitude be
with respect to Ao, or what percentage of the whole pulse must be accounted








for when a reaches 0.1. One way to answer this question is to look at the

values for the existing SHPB systems at the University of Florida and at
a
Tyndall AFB. A short computer code was written to calculate and

Fourier amplitude values for an assumed trapezoidal input based on the
Pochhammer-Chree solutions [32]. It was found (see Appendix B) that for
each system the amplitudes of the Fourier components had fallen to 5-9
a
percent of the original value by the time A reached 0.1. Therefore, a good

minimum amplitude ratio to use for the proposed 6-inch system is about 10
percent. This would assure the one-dimensionality of the response.
In conclusion, because of the form of the equations involved, a pres-
sure bar system can be mathematically scaled up or down to any degree,
without affecting the characteristics of the wave motion. Analyses of the
bars and input pulses used both at Tyndall AFB and at the University of
Florida, showed that a 6-inch system would be equivalent to the existing 2-
and 3-inch systems, if the input pulse duration was roughly 500
microseconds.
The length of the striker bar is related to the pulse duration as
2L
T = where co is the elastic wave speed and Ls is the striker bar length.

The elastic wave speed co for steel is approximately 5080 m/sec. This means
that for a 500 microsecond incident pulse in a steel SHPB, a 1.27 m (50-
inch) long striker bar would be required.


Wave Propagation in an SHPB


Having an understanding of this very important fact, one can now turn
to a more in-depth look at wave propagation in a cylindrical bar, to broaden
the understanding of the overall problem. Three different methodologies
were followed to study this problem. The goal was to compare and contrast








results, understanding the drawbacks, assumptions and reliability of each
one. These three approaches were :

A new numerical method developed by the author using the
Pochhammer-Chree frequency solution and referred to as a
modified Pochhammer-Chree method.
A finite difference numerical simulation of the governing equa-
tions contained in an existing computer code.
Experimental results from the SHPB at Tyndall AFB.
The following sections are a general description of these three approaches.


Modified Pochhammer-Chree Method


As previously discussed, Pochhammer and Chree independently
solved the two-dimensional equations of motion for an infinitely long bar
assuming a sinusoidal waveform. Their solution is what is referred to as the
frequency equation for longitudinal motion of an elastic rod. It turns out to
be a function relating Poisson's ratio, wavelength, and wave speed, and can
be solved for each mode of a cylindrical bar for a particular material. This
solution is neither simple nor explicit. Beginning with the Pochhammer-
Chree frequency equation and assuming only the first mode (MI,1) is ex-

cited, the wave speed of any wave of a given frequency in the bar of interest,
since it is dependent only on its wavelength, can be calculated. Using this
wavespeed and corresponding wavelength along with given material proper-
ties, the longitudinal and radial distribution of displacements and stresses
may be determined from the assumed displacement functions for the "lon-
gitudinal" vibrations. One can now turn to the transient pulse being
propagated in the bar and describe it in terms of Fourier series; that means
representing the waveform by the sum of an infinite number of pulses, each
one having a distinct amplitude, a distinct wavelength and traveling at a
unique speed. This wave speed can be calculated from the frequency equa-
tion. Therefore, each piece of the input pulse can be followed as it travels
down the bar, and at any time, the complete waveform can be reconstructed







numerically by adding up all the pieces. This methodology is currently being
used to correct for dispersion in the SHPB at the University of Florida and
at Tyndall AFB. To carry the logic a bit further, it appears one can still treat
the pulses individually as they encounter the boundary between the
specimen and the incident bar, partially reflect and partially transmit
through, and eventually encounter the second interface where more reflec-
tions and transmissions take place. The basic assumption is that as the
single Fourier component encounters an interface, its amplitude splits ac-
cording to the characteristic impedance values of the two media, (which will
be derived later), but it will maintain its original frequency and velocity. In
other words, each wave component divides each time it encounters an inter-
face, new components are formed each with a different amplitude, but all
with the same frequency. Note also that the sum of these Fourier com-
ponents produce the total incident, reflected, and transmitted pulses whose
wavelengths will also not change.
The implications of this assumption are crucial to the conclusion that
one can theoretically follow each wave component through multiple reflec-
tions, transmissions, and any length of travel, and at any time know its
amplitude and location. With that information one can then reconstruct the
pulse at any time or location and obtain the shape of the response.
A computer program has been written for an SHPB following the logic
just described and is included in Appendix A. A trapezoidal input pulse was
assumed to represent the longitudinal strain at r=a and 2 meters from the
specimen. Fifty-five terms were carried through in the Fourier analysis. At
the onset all the terms in the series are assumed to be in phase and their
relative phases are calculated as they travel down the bar. After multiple
reflections, the pulses are reconstructed at the two strain gage locations on
the incident and transmitter bars (both 1 meter from the specimen). At any
interface between dissimilar materials, velocities and forces must be con-
tinuous. Assuming normal incidence waves only, in a perfectly elastic







medium, then the relationship between the incident, transmitted, and
reflected stresses can be shown to be



at = [2 Alp2 c2/(A2p2 c2 + Alpi ci )] oi


r = [(A2p2 c2 Al p cl)/(A2p2 c2 + Al pi c) ] i


where A is the bar cross-sectional area, p is the density, and c is the
wavespeed. Subscripts 1 and 2 pertain to the two materials, and the wave is
moving from material 1 to material 2. Each term in the Fourier series has a
different wavespeed, and since only Mode MI,i is considered in the analysis,
these wavespeeds are always less than or equal to the infinite elastic speed
VEIp.
For the sake of consistency in comparing the various approaches, this
effort was based entirely upon a compressive SHPB whose incident and
transmitter bars are made of steel, and a work hardened aluminum specimen
of varying dimensions was assumed. Thus, the characteristic impedance pc is
approximately three times greater in the bars than in the specimen. Conse-
quently, an incident compressive wave will be reflected as a tension wave at
the incident bar-specimen interface and the transmitted part will still be in
compression. As this compressive pulse reaches the end of the specimen at
the transmitter bar, it will be both reflected and transmitted in compression
in accordance with the mismatch of characteristic impedances of the bar and
the specimen. This means that a compressive wave is now "trapped" in the
specimen. It loses amplitude at each reflection, but it remains compressive.
It also means that every time the wave reaches the specimen-incident bar
interface it will also transmit into the incident bar as a compressive pulse.
The compressive pulse traveling out of the specimen into the incident bar
superposes itself onto the reflecting tensile part and thus reduces its mag-
nitude. As the compression pulse makes multiple reflections and







transmissions within the specimen, the magnitude of the original tensile
reflection of the incident pulse is reduced by the magnitude of the trans-
mitted compressive pulse from the specimen into the incident bar. The
length of the compressive incident pulse is of finite time, which means the
tensile reflection and the compressive transmission pulses are also of finite
length, and in fact are of identical duration as the incident wave. This
implies the wavelengths of these transient pulses do not change as they cross
interfaces, only their amplitudes are changed.
These phenomena are clearly seen in experimental traces (see for
example Figure 4), and in the Modified Pochhammer-Chree results (see for
example Figure 5) obtained from the program discussed in the previous
page; the reflected tensile wave starts out at a certain value, and then in the
time required to transit twice the specimen length, it jumps to a lower level
due to the arrival of the first portion of the compressive pulse from the
specimen. The amplitudes of the reflected and the transmitted pulses add
up to the amplitude of the incident pulse. In the experiment of Figure 4 the
specimen was plastically deformed, while for Figure 5 the specimen remains
elastic.
It should be mentioned at this point that in order to be able to directly
compare the results, the dimensions of the bars and the locations of the
strain gages were the same in all methodologies. Specifically, since the
Tyndall SHPB is 2 inches in diameter and the gages are 40 inches on either
side of the specimen, both the modified Pochhammer-Chree method and the
finite difference method were run with those values. For the 6-inch SHPB,
the numerical methods assumed strain gages 80 inches on either side of the
specimen.































C-,
So



F:




C-,



1:

I-


edW 'SS381S


o o o o
0 0 0 0
ln in o
I -4






























Z

Tz
I0


II
OE

-I
o
z
rM


o to ro NT O
d dI I I


qtowoooWm6l N
iii iii TY-


S3SS3u.S 03ZnnryVION







As mentioned earlier, the results of the Pochhammer-Chree frequency
equation also lead to the variation of the stresses and displacements in the
radial direction. These variations also depend on the wave speed and the
wavelength of the pulse, and there are an infinite number of solutions each
associated with a particular mode of vibration. After some tedious deriva-
tions [33] the amplitudes W(r) and U(r) of the assumed longitudinal and
radial displacements uz = W(r) f(t,z) and ur = U(r) f(t,z) are


^ Ji (ha) haJo(hr) (1 -f x) /caJo(Kr)
W(r)= iyc +I
ha J J(ha) (x 1) J (Kc a)

J1 (ha) x( ha J (hr) (1 x) 1 /aJ Ji ( r)
U 0 ha (x- )a J, (ha) (x -1) (2x- 1) J1 (K a)


where c = a constant determined by the amplitude of vibration,



(Co) 1-2v

h=y(fpx- 1)V2, K/=y(2x- 1)'/

With this information the radial, shear, and longitudinal stresses can be
calculated as


2 [^J(ha) Jo (hr) a Ji(hr)
=2 ha (1 x) ha1(ha) r J (ha)+


fix 1 Jo (Kr) xfx- la J1(K r)
1 x J1 (K a) 1 x r Ji(/ca)


ur = -2 ip y ch J1 (ha) (J1(ha) J1 ( a) f (t,z)


-2 J(ha) Jo (ha) J1 (K a)
z= -2uyc (ha) [(l+x-fix) haJl(ha) + (1-x)a J( ]f(t,z)







Again assuming only Mode M1,1 is excited, the values of the wave

speed for each component of the Fourier series under consideration can be
calculated. Thus, if the bar is divided radially into a number of increments,
it is possible to calculate the stress and displacement distributions in the bar
at any point at any time. These calculations can be combined with those of
the preceding discussion of the Modified Pochhammer-Chree equation and
the result is a complete description of the elastic wave propagation in an
SHPB.
Care must be taken when solving the equations involving Bessel func-
tions numerically, since as the argument becomes negative, modified Bessel
functions must be used. Specifically


Jo (ix) = Io (x) and Ji (ix) = i II (x),

or in general ,


In (x) = (i)-n Jn (ix)

a
Davies [27] showed that if the ratio is kept below 0.1 then the radial

variation is less than 5%. In an SHPB there are an infinite number of pulses
a
whose W values span from zero all the way to infinity. As discussed earlier,

it is sufficient to make sure that the amplitude of the Fourier components

have fallen to a small fraction of the initial value, when the value of -

reaches and goes beyond 0.1. That is, most of the contributing components
a
of the Fourier series must fall below = 0.1.

A subroutine was added to the Modified Pochhammer-Chree program
to calculate the radial variations for the longitudinal and the radial stresses.
The results of the longitudinal stress for a 2-inch and a 6-inch bar are plotted in
Figure 6. As expected, the lower ( lower values of n), Fourier series terms
















S1-
C' ii /H I
C-0 t o C






I / / / -
I I

J






I- I
of
cu







Ci








Ocn
0e I








UU
LL IP






i0 0o
SLI I I I I
Hn-
1) 011
I II I I I I I I II I I I II I -o
000000000 000000000


SS3I.S IVNIanlIIONO1i 3:ZInVVON








(the ones whose values are low), show a nearly uniform distribution of

a
stress, while as the wavelengths become smaller ( larger ), the radial

variations become progressively greater to a point where they change sign.
If the amplitudes of these higher terms have fallen to a very small value, then
their contribution to the overall pulse will be negligible. Plots of the radial
variations of reflected and transmitted stresses are shown in Appendix A for
a 2 and a 6-inch SHPB (Figures 36 and 37). It is also apparent that as the
diameter of the bar is increased and the input wavelength is kept the same,
the stress will show uniformity for only a few terms. This fact is additional
proof that the one-dimensionality of an SHPB is dependent on the ratio of
the radius to the wavelength, and if kept the same as for the existing SHPBs,
then the wave propagations should be mathematically identical.
Although the higher Fourier terms are highly nonuniform, their con-
tributions are very small. Furthermore, if one normalizes the amplitude
values of the radial and the shear stresses to those of the longitudinal stres-
ses at the center, the results show the radial and shear stresses to be of many
orders of magnitude smaller than the longitudinal values. This can also be
observed in the finite difference calculations, and it is further proof of the
one-dimensional nature of the wave propagation in the bar. Also, as
expected, when the diameter of the bar is increased to 6 inches, the radial
and shear stresses also increase, but remain in the same order of magnitude,
and can thus still be safely neglected.


Finite Difference Numerical Simulation


The two-dimensional equations of motion in cylindrical coordinates
can be solved using an existing hydrocode such as HULL [34]. HULL has a
two-dimensional Lagrangian module that solves conservation of mass,
momentum, and energy simultaneously with the definitions of the deforma-
tion tensor, constitutive relations, and the stress deviators. A centered







finite difference approximation is employed. The topic of constitutive rela-
tions is the most crucial as it includes the relationship between the elastic
and plastic deviatoric strains and stresses, and the relationship between
pressure, density, and internal energy. The accuracy of any hydrocode is
highly dependent on the implementation of the constitutive relations.
In this effort only metals were considered, namely steel and
aluminum. Both are modeled in HULL as elastic/plastic materials, although
the stress levels were kept low enough to keep the steel bars in the elastic
region. Both materials are also considered isotropic. Stresses are decom-
posed into a spherical and a deviatoric component, and the generalized
Hooke's law is used. Yielding is treated following the von Mises criterion,
which assumes that yielding starts when the combined-stress distortion ener-
gy equals the yield value of distorsion energy in a tension or compression
test. Specifically, it relates the second deviatoric stress invariant to a
specified yield function. Once yield is reached, the deviatoric stresses are
corrected and reset to be normal to the yield surface in stress space. The
total stresses can then be calculated. Using HULL, a trapezoidal pressure
pulse was input on the incident bar and the specimen was specified as an
elastic/strain-hardening material. The HULL output stations included the
locations of the strain gages on the bars, plus one within the specimen itself.
Sliding interfaces with no friction were used between the two materials for
most calculations. To study the effects of friction, locked interfaces that
allow no sliding to occur were used. The bars are made of steel and remain
elastic at all times. The first runs were made to model as closely as possible
the experimental data available for aluminum from the Tyndall AFB SHPB.
Appendix D contains a HULL sample input file.
HULL was also run to investigate the complicated wave interaction
phenomena at the specimen-bar interfaces, especially in cases when the
specimen is smaller in diameter than the bars. The next step was to study the
effects of a 6-inch diameter bar, with two input pulses. One input pulse with
a wavelength long enough to give a one-dimensional response was used.







Additionally, one with a much shorter wavelength was used to show how the
stresses and displacements vary in the radial direction, and disperse after a
short travel, making the conventional SHPB analysis invalid.


Experimental Method


How does one know how good or even reliable the HULL output is?
In order to answer this question, a direct comparison was sought between
existing experimental data obtained at Tyndall AFB, and the calculated
results from the HULL code. Specifically, numerous compression tests were
run on aluminum 6061-T6511 specimens between steel incident and trans-
mitter bars. The input pressure was varied to assess strain rate sensitivity,
and stresses and strains were recorded at strain gages located in the incident
and transmitter bars. It was found that (as already known) this particular
aluminum is strain rate independent; that is, the stress strain curve is not
affected by the rate at which the load is applied. This is very important
because the HULL code does not take strain rate into account. Therefore, a
comparison will be meaningful between the experimental data and the com-
puter results for an aluminum specimen. The available data was obtained
from a 2-inch diameter SHPB and varying specimen sizes. HULL was then
run with a new material added to the material library to match as closely as
possible the 6061-T6511 aluminum properties.
In particular, aluminum is modeled as an elastic/plastic material, and
one would expect to observe that stress strain relationship in the output.
One should be able to recreate the relationship between stress and strain
since that is what the SHPB is designed and used for. The rationale was to
prove the validity of the numerical tool by comparing existing data with the
results of the code, and then move on to the proposed 6-inch bar and have
the confidence to trust and draw conclusions from those results.







Comparison of Results


A number of compressive tests were conducted at Tyndall AFB using
a 2- inch SHPB on aluminum specimens. The diameter and length of the
specimen, as well as the input pressure were varied to assess their influence.
The results for three cases are shown in Figures 7, 8, and 9. To compare
experimental and numerical results, HULL was run matching as closely as
possible the conditions in the Tyndall tests. One needs to realize however,
that no matter how good the numerical model is, there are some parameters
one cannot predict or reproduce. For example, the input pulse is not in
reality a perfect square pulse, nor is it perfectly uniform along the cross
section as numerically modeled. Furthermore, the constitutive equations
for steel and especially for aluminum (which goes beyond yield in the course
of the test), are theoretical models and by no means exact. Finally, the
effects of friction are not included in the numerical scheme.
Having said that, one is then looking for a qualitative match in the
data, more so than exact values of stresses or strains. What this means is
that certain trends that have been observed experimentally should show up
numerically. For example, as the specimen gets smaller compared to the
incident bar, experimental results show that the transmitted stress also be-
comes smaller. Consequently, the reflected stress increases and its shape
becomes more uniform as the initial high peak starts to blend in with the rest
of the reflected pulse. The HULL calculations also show these phenomena.
Although the actual amplitude values do not match, the relative change
between tests does. In general, the experimental data gives higher trans-
mitted stresses, but it does so consistently. Experimental results are shown
in Figures 7, 8, and 9 and can be compared to the numerical results from
HULL shown in Figures 10, 11, and 12. The results of the HULL calculations
show very good agreement with the experiments, which point out the validity
of the HULL code.






















Ei



















00
I-
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'0















rij
4O






-!
ce
C 0





4"S





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o o o


0 o
4 1


edW 'SS3UIS











































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o
o

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o o o a
o o o0 0
0 0 0 0

I I


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ct2
6














r-











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-j
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a 03














o
ia a







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S: o .













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a E
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cu
a iu




















Cu


RdW 'SS3u.LS



















2.01E+00

1.6gE+00

1.21E+00

8.01E-01

4.01E-01

0.0

-4.01E-01

-8.01E-01

-1.21E+00

-1.61E+00

-2.01E+00
0.00



1.00E-03

8.00E-04

6.00E-04

4.00E-04

2.00E-04

00

-2.00E-04

-4.00E-04

-6.0OE-04

-8.00E-04

-1.00E-03 -
0.00


0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.1
T (MSEC)
SPUT HOPKINSON BAR 0.4 CM ZONING Problem 4.0000


SSTA= 1 XO= 2.33 YO= 99.80
MAX= 5.372E-04 MIN=-8.220E-04


0.11 0.22 0.33 0.44 0.55 0.66 0.77 0,88 0.99 1.1
T (MSEC)


Figure 10. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY)
Pulses for an Aluminum Specimen, Diameter = 3.18 cm,
Length = 3.18 cm


STA= 1 XO= 2.33 YO= 99.80
MAX= 1.287E+00 MIN=-1.969E+00



















2.01E+00

1.61E+00

1.21E+00

8.01E-01

4.01E-01

0.0

-4.01E-01

-8.01E-01

-1.21E+00

-1.61E+00

-2.01E+00
0.00



1.00E-03

8.00E-04

6.00E-04

4 00E-04

2.00E-04

0.0

-2.00E-04

-4.00E-04

-6.00E-04

-8.00E-04

-1.00E-03
0.00


0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.1
T (MSEC)
SPUT HOPKINSON BAR 0.4 CM ZONING Problem 3.0000


STA= 1 XO= 2.33 YO= 99.80
I MAX= 6.267E-04 MIN--8.215E-04


0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99
T (MSEC)


Figure 11. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY)
Pulses for an Aluminum Specimen, Diameter = 2.54 cm,
Length = 2.54 cm


STA= 1 XO= 2.33 YO= 99.80
MAX= 1.500E+00 MIN=-1.968E+00



















2.01E+00

1.61E+00

1.21E+00

8.01E-01

4.01E-01

0.0

-4.01E-01

-8.01E-01

- 1.21E+00

-1.61E+00

-2.01E+00
0.00



1 00E-03

8 OOE-04 *
8.00E-04

4 00E-04
4.00E-04

2.00E-04

0.0

-2.00E-04

-4 00E-04

-6.00E-04

-8.00E-04

-1.00E-03 -
0.00


STA= 1 X0= 2.33 YO= 99.80
MAX= 1.666E+00 MIN=-1.973E+00



















0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.'
T (MSEC)
SPLIT HOPKINSON BAR 0.4 CM ZONING Problem 6.0000


STA= 1 XO= 2.33 YO= 99.80
MAX= 6.938E-04 MIN--8.238E-04


0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88
T (MSEC)


0.99 1.1


Figure 12. HULL Output. 2-inch SHPB. Stress (TYY) and Strain (EYY)
Pulses for an Aluminum Specimen, Diameter = 0.56 cm,
Length = 0.56 cm







It makes sense physically that as the specimen gets smaller, more and more
of the incident pulse is reflected back, to the limit case when there is no
specimen and the reflected trace is identical and reverse of the incident. It
also seems reasonable that when the specimen diameter is less then half of
the SHPB diameter, little of the incident pulse gets transmitted into the
specimen, which means that even less of this transmitted pulse will get back
to the reflected pulse and therefore its basic shape will not be changed.
Hence the appearance of a distinct peak in the reflected trace is reduced
when the specimen is small.
It has already been shown in Figure 5 that the Modified Pochhammer-
Chree method also shows the characteristic "peak" in the reflected trace
even though the solution is elastic. The only difference between an elastic
and an elastic-plastic specimen is in the magnitude of the stress that may be
applied. Therefore, the only difference in the shape of the traces in the
modified Pochhammer-Chree method will be a higher compressive reflec-
tion out of the specimen, resulting in a lower reflective plateau after the
initial "peak", than for the corresponding non-elastic case. To show this,
three experimental tests were run with the same specimen configuration at
different levels of stress; one level was kept in the aluminum elastic region,
the other two exceeded the yield value for aluminum. Figure 13 shows how
after the initial peak in the elastic case (top figure), the reflected pulse
almost goes down to zero, while as the input stress is increased (middle and
bottom figures), the plateau after the peak increases also.
Another issue to be explored experimentally and numerically is the
effect of decreasing the pulse duration with respect to the diameter of the
bar. As discussed, the equations governing an SHPB show that increasing

the ratio will affect two important things. The first thing is that the pulse

will disperse more; that is, the contribution from the slower traveling com-
ponents will be greater, and the overall waveform will distort and spread out
more as it travels. The second thing is that the wave propagation will be less one
dimensional in nature, which means that the longitudinal stress will have















<-REFLECTED


0.0


o1 <-INCIDENT -TRANSMITTED

50 -25.0
i--
In



-50.0 -


0.0 250.Ous 500.Ous 750.Ous
TIME




<-REFLECTED



0.0



i- -50.0 -

/<-TRANSMITTED
<-INCIDENT
-100.0 --

I I I I I I I I I I I I
0.09 250.Ous 500.0us 750.0us
TIME
I T r n I m I I I I I I I I I _
100.0 -




-.-REFE CTED




U)










TIME



Figure 13. Experimental Traces at Three Incident Stress Levels (60 MPa-Top,
S a-iddle, 167 MPa-Bottom). Specimen Diameter 1.253


Length = .625 in, 2-inch S B
0.05 250.0us S00,us 750.0us


Figure 13. Experimental Traces at Three Incident Stress Levels (60 MPa-Top,
100 MPa-Middle, 167 MPa-Bottom). Specimen Diameter 1.253 in.,
Length = .625 in., 2-inch SHPB








larger variations along the radius and the radial and shear stresses will be
a
less negligible as increases. Numerically one can observe the effects of

decreasing the input wavelength by using HULL and the Modified Pochham-
mer-Chree method. The following runs shown in Table 1, were made to test
the methodologies against what it is analytically believed to be true.


Table 1. HULL and Modified Pochhammer-Chree Runs



DIAMETER WAVELENGTH a FIGURES
(inches) microsecondss) A

2 250 1/125 3, 14

6 750 1/125 15


6 500 3/250 16, 17


6 250 3/125 18,19


Figures 5, 15, 16, and 18 show results from the Modified Pochhammer-
Chree program, while Figures 14, 17, and 19 are HULL output traces.
Notice that HULL appears to have two different coordinate systems from
one figure to the next. The reason for this is that the runs were made on
different computer systems (VAX and CRAY) and each system has a slightly
different version of HULL. In one z is the longitudinal coordinate, and in
the other, y is the longitudinal coordinate. This discrepancy should not
a
cause confusion. It can be seen that the pulses disperse more when s is greater.

a
The results also confirm that if X is the same, regardless of the bar

















2.41E+00
STA= 1 XO= 2.33 YO= 99.80
1.81E+00 MAX= 1.743E+00 MIN=-3.323E+00

1.21E+00

6.01E-01

S0.0

-6.01E-01

S-1.21E+00-

-1.81E+00

-2.41E+00

-3.01E+00

-3.61E+00
0.00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.88 0.99 1.1
T (MSEC)
SPUT HOPKINSON BAR 0.4 CM ZONING Problem 5.0000

1.50E-03
STA= 1 XO= 2.33 YO= 99.80
1 20E-03 MAX= 7.268E-04 MIN--1.386E-03

9.00E-04-


3.00E-04


0.0

-3.00E-04

-6.00E-04

-9.00E-04

-1 20E-03

-1 50E-03
0.00 0.11 0.22 0.33 0.44 0.55 0.66 0.77 0.8B 0.99 1.1
T (MSEC)












Figure 14. HULL Output. 2-inch SHPB. 2-inch Diameter and Length,
Aluminum Specimen, 257psec Pulse












































































o d000o dd000o d-0 -
I I I I I I l I I I I


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0

YC
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a
w Oi




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0I 0





























I I I I I I I I 1 I
I

0 0 0 0 0^ 8
--i

s r
j>o
"^ ^

^ '
n^nr-o Sc~c
o

SSMuLS iaMGIONI O3ZrWNON













































T (MSEC)


Figure 17. HULL Output. = 500 psec Pulse, Station 1 (Top)
Bar Center, Station 2 (Bottom) Bar Surface





























I


O2 c



0 0 E I
0.
Z<






CD W

00



z z N
E
















I I I I I I I I I I I I I I I I 0
LI




u4)

N7
-P

4n







I l I I I TI I


S3SS3ylS a3ZnVMviON
































I ~ : II

1+ 0







o Bo T ai
00
x =0




N LV




N 00





t 0







I


Cb
0
W 4






X 0 0 0 0 0 0
e'4 0 0 oi N
N ID I I


(B)) ssais








diameter, the waveforms are mathematically the same; this is shown in
Figures 5 and 15. Figures 18 and 19 indicate that an input pulse of 250
microseconds for a 6-inch SHPB is not recommended. It appears dispersion
becomes a problem, even after a very short time, and also the one-dimen-
sionality of the motion is questionable. Experimentally, one can achieve
different wavelengths by changing the length of the striker bar. The result-
ing waveforms for a 16-inch and a 25-inch striker bar are shown in Figure 20
( all three bar locations are on the incident bar ). Notice the more
pronounced dispersion for the shorter wavelength.
As discussed, along with more wave dispersion, the one dimen-

sionality of the motion in a bar is also affected by the change of A values.
a
Again, one can turn to the two numerical schemes and observe how, as -

increases, the radial and shear stresses increase, and the longitudinal stress
shows more variations along the radial direction. Figures 21, and 22 show all
stress components on the surface of the incident bar at the strain gage
location for a 2-inch bar and a 250 microsecond pulse, and a 6-inch bar with
a 500 microsecond pulse. Incident and reflected stresses are shown. Al-
though the radial and shear components of the stresses are higher for higher

values of -, they nevertheless remain negligible when compared to the

longitudinal stress. Figures 23, 24, and 25 show the pulses at the center and
at the surface of the transmitter bar at the strain gage location. Figure 23 is
the result of the modified Pochhammer-Chree method for a 2-inch bar and
250 microseconds. Figures 24 and 25 are HULL output for 250 and 500
microsecond input pulses respectively. Figure 23 shows a difference of
about 1.5% between the two curves. The variation in stress along the cross
section is roughly 5% for Figure 24 and 2.5% for Figure 25. Therefore, the
a
uniformity of the longitudinal stress is degraded as A is increased, although

for the proposed 6-inch bar and a 500 microseconds pulse, the wave motion
should be very close to one-dimensional.










distances from striker bar end


0.0001 0.0003 0.0005 0.0007
Time (sec)


0.0001 0.0003 0.0005 0.0007
Time (sec)


Figure 20. Experimental Traces at Three Incident Bar Locations for Two Striker Bar
Lengths. 16-inch (Top) and 25-inch (Bottom), 2-inch SHPB


-1 --I
-0.0001


-1 -I-
-0.0001


0.0009


0.0009




















2.41E+00



1.81E+00



1.21E+00



6.01E-01



0.0



-6.01E-01



-1.21E+00



-1.81E+00
II

-2.41E+00



-3.01E+00



-3.61E+00 -
0.00


REFLECTED


STA- 1 XO= 2.33 YO= 99.80













7- 7 YY


DENT ---*


0.11 0.22 0.33 0.44 0.55
T (MSEC)


0.66 0.77 0.8B 0.99 1.1


Figure 21. HULL Output 2-inch Bar, 250 usec Pulse, All Stress Components
on the Surface at Incident Bar Strain Gage

















































0.75
T (MSEC)


Figure 22. HULL Output. 6-inch Bar, 500 ,sec Pulse, All Stress Components
on the Surface at Incident Bar Strain Gage


























Wm I
uJm




-LU

N o





S I I I I I I I I I I I I I 0
LLJ





I- o-' d^ 1 c o--





Io 66I I I I I -
I ill iiI
-I r 2 -- \ i ^----- *
^~~~~~ ~~ rM LC MK ^ O ior ac -i iK
o d d d d l ^ -^


S3SS3I1S 1N3aIONI a3ZInVI-NON










3.00x 10-'
STA= 4 XO = 7.410E+00 MAX = 2173E-01
YO 4.998F-'.; -,: ",:"'-,

o.oo "
03.0010




-6.00x10"


-9.DO00xD'


-1.20


-1.50


-1.80

Z
-2.10


-2.40


-2.70
0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 '.20 1.35 15
TIME (MSEC)




3.00x 0-'
O.03.00 x 10-' YO-----------4-----------------------------------------------
STA= 5 XO = 1.900E-01 MAX 2 69 l1
YO 4.99BE+02 MI --2. 1 0

0.00
Z




-6.00x I '



-9.00x1 0'
-9.00 ID"


-1.20
Z
LA 2
-1.50






-2.10
-1.80 L







HP = "hoop" stress I
-2.40 2 -------- i -------- i -------- i -------- --------
-2,70
0.00 0.15 0.30 0.45 0.60 0.75 0..90 1.05 1.20 1.35 1.
TIME (MSEC)




Figure 24. HULL Output 6-inch Bar, 250 ,sec Pulse.
Surface (Top) and Center (Bottom) Transmitted Stresses


















1.21E+00

8.01E-01

4.01E-01

0.0

-4.01E-01

-8.01E-01

-1.21E+00

-1.61E+00

-2.01E+00

-2.41E+00

-2.81E+00
0.00


1.21E+00

8.01E-01

4.01E-01

0.0

-4.01E-01

-8.01E-01

-1.21E+00

-1.61E+00

-2.01E+00

-2.41E+00

-2.81E+00
0.00


0.15 0.30 0.45 0.60 0.75
T (MSEC)


0.15 0.30 0.45 .60 0.75 0.90 1.05 1.20 1.35 1.5
T (MSEC)


Figure 25. HULL Output 6-inch Bar, 500 ,sec Pulse.
Surface (Top) and Center (Bottom) Transmitted Stresses


STA= 3 XO= 7.41 YO= 399.80
MAX= 6.370E-01 MIN=-2.442E+00


STA- 4 X0=
MAX- 5.477E-01


0.90 1.05 1.20 1.35 1.








A tool like HULL offers the advantage of being able to "look" at the wave
response, stresses, accelerations, etc., anywhere in the bar. In particular,
two questions regarding the operation of an SHPB have been of interest for
some time. The first one has to do with the initial oscillations that are
observed experimentally and whether they are due to something inherent to
the equations of motion, or whether they are the result of some experimental
characteristic like friction, or striker misalignment. The second question
has to do with what really goes on at the interfaces between the specimen
and the bars, especially when the specimen is smaller in diameter. To
answer the first question one can look at the HULL stress output at points
very close to the impacted ends. Figures 26 through 28 show curves for
various diameter bars and various input pulses at two locations on the inci-
dent bar, one near the impacted end, and another one meter from it. The
a
following observations can be made; in the cases where A is larger, the

oscillations in the input pulse are more pronounced and they take longer to
damp out. Further, if one compares these results with the experimental
traces (Figure 20), the similarity in the response is striking. Since the finite
difference calculation does not take into account friction, misalignments,
etc., it can be deduced that the so called "end effects" are due to inherent
characteristics of the wave motion not to the way the experiment is carried
out. That means higher frequency modes are excited at the impacted end of
a
the bar. If one refers back to Figure 3, the cases with larger show more

oscillations because the response is further to the right on the x-axis, and
thus more modes of vibrations are possible. Nevertheless, as predicted by
Davies [27] and others, these high frequency modes damp out quickly, so
that after about ten diameter lengths the only mode of vibration present is
mode M1,1.
The second question presents more of a problem; the wave interac-
tion at the bar-specimen interfaces is extremely complicated, especially if the
specimen yields at some point during the process. Looking at the stress

























a










co






a,
I-























c=._e
C.4
o o
i- s







i3J


oo o o o o o o o o a
x X X X X xxx xx
x X X X 0 0 0 0 0 x O O O x 0 0 0 0 0
0o 0o 0 0 0 0 6 o o 0 0 0 0 'o

(33/3NNO) ZZi (33/3cNa) ZZL













1 .0 STA= 1 XO = 7 410E+00 MAX = 9.756E-0
YO 5.000E+00 MIN --3 328E+0(

1.00


5.00' 10-'
0.00





-5 .00 0 '



-100
2 z z


-1.50






2.50 z


-2.00
z
z HP "hoop" stress a


0.00 0.i5 .30 045 0.60 0.75 0D 1.05 1 20 135
TIME (MSEC)
1.50
STA= 2 XO = 1.900E-01 MAX 9.
YO 3.002E+02 MIN 3.

1.00



5.00x 10- -



0.00



-5.00x 10'



- 1 .00



-1.50 Z



-2.00



-2.50







-3.50
0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.3
TIME (MSEC)





Figure 27. HULL Output. 6-inch Bar, All Stresses Near Impacted End (Top)

and at the Strain Gage Location (Bottom)






64









1 50 Sl X0 2 32tO MAx 1 217E00
YO 5000E+00 MIN ---3 25E 0N





5.00 10 p 1Z






S00
5 00O '1







- 150


-2. 00
r
z
2 50


-J.oo ;z
HP = "hoop" stress -= o

-3 s0 000 0a15 030 0 45 060 0.75 090 1,0 1 20 135 I 5
TIME (MSEC)
SPLIT HOPKINSON BAR LOCKED NODES Problem 5 1000

3 50
3.50 I 2 xO 2 .32E+00 MAX 2 474E+00
O 1 002E402 MIN --3 323E 00

2.80




















-1 4( z
7 0C. 10-'














0 00 0 15 0 0 0 45 060 0.75 090 1.05 1 20 1 5 I
TIME (MSEC)





Figure 28. HULL Output. 2-inch Bar, All Stresses Near the Impacted End (Top)

and at the Strain Gage Location (Bottom)







components along one such interface from a HULL calculation, it is very
difficult to make any definite statement. To help in the understanding of the
complicated stress wave interaction, Figure 29 shows the important features
and magnitudes of all stress components along the incident bar-specimen
interface. The longitudinal stress (uniform and in compression, of mag-
nitude 167 MPa sometime prior to this interface), at the interface varies
from almost zero at the bar surface to 100 to 80 MPa at the center. Shear
and radial stresses become of the same order of magnitude as the lon-
gitudinal stress. What is worth noticing is that at the center aeo or the "hoop"
stress is very large, in fact larger than the input stress. Also at this point the
displacements in the longitudinal and radial direction are small. It appears
then, that shear and circumferential stresses play an important role at the
edge where the specimen ends against the bar face. In a sense, the incoming
stress has to "squeeze" itself through a smaller cylinder therefore it is not
surprising to find other than longitudinal stresses at the interface. What is
remarkable is how quickly it recovers to an almost uniform state of stress
within the specimen only one centimeter from the interface (Figure 30), and
in the transmitter bar at the strain gage location (Figure 31), where the
longitudinal stress is the only non zero component. Figures 30 and 31 cor-
respond to the same SHPB as in Figure 29. Notice that a00 in Figure 30 is
only about 50 MPa while it peaked at 220 MPa at the interface; that amounts
to a reduction of 77% in one centimeter. Appendix C contains more HULL
output traces which can further explain the above discussion.
Finally, Bertholf and Karnes [23] pointed out that the first interface
between the bar and the specimen effectively filters out most of the oscilla-
tions in the incident pulse, and thus in the cases when these oscillations are
large for one reason or another, it is wise to use only the transmitted stress
to calculate the response in the specimen. The analytical tools also show
this filtering effect of the specimen, and it appears the use of the transmitted
pulse only, may be a worthwhile alternative for the proposed 6-inch SHPB.







































0 6
soo
































-2l01E-0.
001E-





S-tcE-or












-S 0E-Of
-0I-0'o
O0








- OI4-01

-40 E-on


wow-
sor-ol
2 076-of






I 00






-0 t

o6 01 O 0:2 O. 0 0o** 0 s o00 o 0.8 000
T (inco



Figure 29. HULL Output 2-inch Bar, All Stresses at Three Locations

at the Bar-Specimen Interface


STATION 1


STATION LOCATIONS







3








SPECIMEN





























t STATION 3






I H.0 \


I ot 0oi oU 0* o aO or o 0 0 i
T acm)


STATION 2


ooo~~~~~~~ ~ 05621aJ11 ss ooa o

























































































HULL Output 2-inch Bar, Stresses in Aluminum Specimen at Two
Locations Near the Surface (Top), Near the Center (Bottom).
Compare with Figure 29


9 00x10 '


2.00


1.50





5.00x10-








-1.00


-1.50


-2.00


-250


-3.00
0.






Figure 30.










3,0010* 5A 3 X 2.328a*00 WX 2 774E+08
Y 3.000E+02 MIN ---4 50E+SO











-.80. 10















-s50.to'


-540o000 0o1 030 0o45 0.60 0.75 0.90 .0 1.20 1,35
TnE (.EC)


3 0010,
















2 70., '
-2 TO"ID*


-3600-'


4 lD'


E r








- 00oo 0'



I 01.10'


-2 70l0'


-360 l00 "


-454010 -'


Figure 31. HULL Output. 2-inch Bar, Transmitted Stresses at Three Radial Locations.

Surface (Top), Middle (Middle), Center (Bottom).

Compare with Figure 29







The two analytical methodologies used have advantages and disadvantages,
and the use of one over another depends on the particular need and applica-
tion. The finite difference solution is the most accurate if the constitutive
relationships of the materials involved are adequate. It is also the most time
consuming and expensive. The Modified Pochhammer-Chree method
developed in this study gives a good qualitative look at the wave propagation
in a typical SHPB in a very short time. It can be reliably used to assess the
one-dimensionality of the elastic wave motion, as well as dispersion effects
and cross sectional variations and values of all the stress components. It is a
tool that can also be used to investigate if a particular experimental setup is
affected by friction or inertia by predicting what would happen in the ab-
sence of friction and inertia. Furthermore, a typical HULL calculation took
anywhere between three and six central processing unit (CPU) hours on a
CRAY Y-MP computer at an average cost of $800.00 per job! The modified
Pochhammer-Chree method takes seconds to run on most machines at an
average cost of less than $2.00. Although the Modified Pochhammer-Chree
method assumes a purely elastic response, it does not affect its usefulness
since one would not use any of these analytical tools to investigate true
material response, but rather to study the reliability of a certain SHPB
configuration. These numerical codes are not intended to replace the ex-
perimental method, but rather to study the validity of the data reduction, the
implications of a geometric change, the effects of a different input pulse, and
so on. After all, if the true constitutive relations of the specimen material
were known one would not need to study its response with an SHPB. The
Modified Pochhammer-Chree method can give the results it was designed to
give with any elastic specimen, as long as the characteristic impedance of the
material and the bars are known.







Inertia and Friction Effects


The analytical methods discussed are useful tools to study the wave
propagation of any SHPB, and within their limitations and assumptions can
assess the one dimensionality and therefore the usefulness of different bar
sizes and configurations. However, the reliability of an SHPB can be
influenced by radial and axial inertia and by friction between the specimen
and the elastic bars. Neither the modified Pochhammer-Chree method nor
the finite difference code HULL take those into account. Approximate cor-
rections have been derived to account for the effects of radial and axial
inertia [7] and friction [8]. The objective of this investigation is to under-
stand when and how inertia and friction become important factors and
whether the SHPB diameter is a key parameter on inertia and/or friction
contributions. In other words, one would like to know whether increasing
the diameter of a SHPB will increase the effects of inertia or the effects of
friction.
Inertia effects become more pronounced as either the initial stress in
the input bar becomes high, or the strain rate is increased [24]. The former
causes very large stress and strain gradients which from the equations of
motion, make the inertia term become important [35]. High strain-rates do
not appear to cause inertia effects per se, rather the way the higher strain-
rates are achieved can cause inertia effects to become important. Specifical-
ly, if one increases the input stress to obtain a higher strain-rate, and this
input stress is in the form of a step function, then apparently the combina-
tion of high input stress, and the oscillations caused by radial waves resulting
from the short rise time of the input, causes inertia effects to become quite
important. Bertholf and Karnes [24] eliminated one of these two causes by,
numerically, using a ramp input function with a much longer rise time. Their
results show that doing this effectively eliminates the inertia effects, even
though the same levels of strain rates were achieved. This means that the
magnitude of the input stress is not a major cause of inertia effects, rather







the short rise time associated with the high input seems to significantly
contribute to inertia effects. Therefore, since in an SHPB one is limited to a
step input pressure, there appears to be a maximum achievable strain rate
beyond which the validity of the experimental data becomes questionable.
Bertholf and Karnes calculated a maximum allowable strain rate for their
SHPB. They claim that since their numerical calculations are based on
constitutive equations which are rate independent, their results can be ap-
plied directly to different diameter systems. For a 1-inch diameter bar, a
length L over diameter d (L/d) = 0.3 in the specimen, and a step input stress,
Bertholf and Karnes calculated a maximum strain rate of 400 1/sec. For a
ramp input, this value is much higher for the reasons discussed earlier. If
the diameter is increased by a factor n, then the new maximum strain rate is
400/n 1/sec. This scaling cannot be applied directly to the proposed 6-inch
bar because the L/d of the specimen will not necessarily be 0.3. In fact it will
be about one for concrete specimens to ensure the same aggregate density in
all directions. Therefore one needs to go back to the equations that Davies
and Hunter [7] derived to calculate the contribution of inertia based on the
kinetic energy due to both axial and radial motion. Their result is



(L2 2 d2 d'e (19)
Us = Ub+Ps(Vs (1



where Ub is the measured stress, subscript s pertains to the specimen and
L2 d2
Sand v--are the axial and radial inertia contributions respectively.
6 8
Equation (19) can be rewritten in nondimensional form as



S- ab d2 E/d t2 ps d2 L L (20)
Ub Ub 6 Ld J








The left hand side is the error in the measured stress. If one assumes
a nominal Poisson's ratio of 0.3, the contribution of the term L/d can be
calculated relative to Bertholf and Karnes L/d of 0.3. This leads to the
conclusion that for a nominal L/d = 1.0, the error in the measured stress will
increase by a factor of 40. This means that the maximum strain-rate they
calculated should be reduced even further. This would lead to unacceptably
low strain-rates. Therefore, the thing to do is accept the errors due to
inertia effects, and correct them using equation (19) in the data reduction
algorithm.
Furthermore, if one looks at typical strain rate plots from actual
SHPB experiments, it appears that in most cases the strain rate very quickly
levels off to some constant value. What that means is that the derivative of
the strain rate will eventually approach zero, and it may be that the inertia
effects will not be as important after the initial 100 or so microseconds.
Friction presents more of a problem in the sense that its effects are
not easily quantifiable or easy to predict. It is shown from numerical cal-
culations that the measured stress will be larger than the true stress, result-
ing in an artificially high strength in the material being analyzed. But how
much? Early work and studies done by Davies and Hunter [7] and by Saman-
ta [10] show that in general, if the diameter and the length of the specimen
are of the same order of magnitude, then friction contributes to only a few
percent error. It appears that the important parameter is the ratio of the
area constrained by friction over the volume of material in the specimen. In
other words, a thin specimen will be more affected by friction than a thick
one. Since the area constrained by friction is d2/2 (two faces) and the
volume is n d2 L/4 then the key parameter in determining friction effects
is 2/L. The larger this value the greater the error due to friction. This
means that increasing the diameter of the SHPB bars and specimen will not
worsen friction effects. Although an exact value for friction effects is not
obtainable, one can assess the order of magnitude of the error by running
HULL with "locked" interfaces between the bars and the specimen, and then







compare the results with the other runs where the interfaces were friction-
less. (There is no option in HULL for modeling friction). The results then
bound the friction problem as the "locked" case represents a worst case
scenario. Figures 32 and 33 show the transmitted stress for a 2-inch bar with
zero and infinite friction, respectively. Figures 34 and 35 show the same
thing for a 6-inch bar. As expected, the cases with infinite friction show a
higher transmitted stress, implying an artificially higher strength in the
specimen. It is interesting to notice that the error due to friction in the
2-inch bar is about 16.4% and in the 6-inch bar is only 5.6%. This agrees
with the criterion derived above which states that the error is proportional
to 2/L. It appears the larger SHPB set up will be less susceptible to friction
in light of the fact that the specimen length will be increased along with the
diameter. However, one should not imply from this that friction introduces
errors in the existing systems of the order of 10-20%. That figure is an upper
bound for a case where the specimen is attached to the bars. In reality, the
friction coefficient will be somewhere less then 1.0, especially if care is
taken in lubricating the ends. Therefore, the errors due to friction are
negligible in typical SHPBs and will be so also for a proposed 6-inch system.
If, however, one was to test thin specimens, then the data reduction should
take friction effects into account.


















2.51E+00



2.01E+00



1.51E+00



1.01E+00



5.01E-01



0.0


-5.01E-01



-1.01E+00



-1.51E+00



-2.01E+00



-2.51E+00
0.00


0.11 0.22 0.33 0.44 0.55
T (MSEC)


0.66 0.77 0.88 0.99 1.1


Figure 32. HULL Output. 2-inch Bar, Transmitted Stress with Zero Friction
Between Specimens and SHPB


STA= 3 XO= 2.33 YO= 300.08








































N
SI


o _
>jw o a

in

F

i 0 w











N o0
0









0





ox 0 0 0 0
*N 0a



( ) ss
(GN) SS3H.S























































0.75
T (MSEC)


Figure 34. HULL Output. 6-inch Bar, Transmitted Stress with Zero Friction













































































T b b
0 -
X 0 0 0
o 0 0 o 0
6 0 I I I


77












0





"- --- -- ^
o












IN N

N







0




























o 00





I *I
(e)





0



-- i -- -- ] --6-- -- .0













CHAPTER 4


SIX-INCH SHPB FEASIBILITY


Summary


The objective of this effort was to assess the feasibility of a 6-inch
SHPB. There is an abundance of experimental data for existing SHPB sys-
tems, and it was used in this study to test the accuracy and reliability of the
numerical finite difference code HULL.
The first part of this work researched the multitude of efforts con-
ducted on the subject in the past years. In particular, it was established that
it was important to look at the two-dimensional equations of motion as the
simplified one-dimensional case may not be adequate for a larger system.
The Pochhammer-Chree solution for an infinitely long bar was used as the
baseline for the mathematical analysis. Early papers on inertia and friction
effects were studied, and the methodology extended to the 6-inch bar sys-
tem, primarily to have a feel for the magnitude of the errors introduced by
those factors.
Taking a closer look at the governing equations and at the Pochham-
mer-Chree frequency equation, it became apparent that the problem of wave
propagation in a cylindrical bar is independent of scale if the ratio of the
diameter over the wavelength of the traveling pulse is kept constant. This is
not a new idea, but the implications in this study are crucial. It means that
in theory at least, an SHPB can be scaled up or down as long as the input
pulse is also scaled up or down. It does not mean the SHPB response is one
dimensional, rather it implies that since the simple one-dimensional analysis







has been proven reliable for the existing systems, it should apply just as well
to the proposed 6-inch SHPB.
The next part of this effort looked at three methodologies in order to
compare and contrast the results. The goal was to have a firm understanding
of the implications of increasing the size of an SHPB. The first two included
a finite difference numerical technique, and experimental traces obtained at
Tyndall AFB. The third method, called the Modified Pochhammer-Chree
method for elastic systems, consisted of computationally following each
Fourier component of the input pulse as it reflects and transmits at the
bar-specimen interfaces. Each pulse has a unique velocity and wavelength,
which enables one to calculate its position in time. The complete pulse can
then be reconstructed at any time. Furthermore, the distribution of dis-
placements and stresses along the radial direction can also be calculated if
the wavespeeds of the Fourier components are known. It was shown how
this methodology gives surprisingly good results, considering the many or-
ders of magnitude difference in computing time compared to the finite
difference technique.
All three methodologies reaffirmed the initial statement of scale in-
variance of any SHPB. It was shown how if the input pressure pulse has a
wavelength of 500 microseconds, a 6-inch SHPB will behave as a one-dimen-
sional wave propagation system, and thus all the current data reduction
schemes and the current assumptions will be valid.
Finally, the effects of friction and inertia were analyzed. It turns out
friction causes only a few percent error, even for a large 6-inch bar. Inertia
however, will affect the results of an SHPB, and must be accounted for in the
data analysis of a new 6-inch system. Inertia effects increase as strain-rate
increases.







Conclusions and Recommendations


The following conclusions can be drawn from this analytical study.


1. A 6-inch SHPB is theoretically feasible, and in fact identical to any
existing system if the input pressure pulse duration is scaled appropriately.
Specifically, a 500 microsecond wavelength is recommended.


2. The modified Pochhammer-Chree method can be used to assess the
validity and study the wave propagation in any size SHPB. The results give
the incident, transmitted and reflected pulses for different diameters,
wavelengths and materials. This method can be applied to any input pres-
sure if it can be described by a Fourier series.


3. If the specimen length and diameter are kept in the same order of
magnitude, one need not worry about friction in the proposed 6-inch SHPB.
It will cause only a small error in the analysis.


4. The new proposed 6-inch SHPB will need a correction in the data
analysis to account for inertia. This is true especially if the input pulse is a
step function and if strain rates higher then 10-20 1/sec are desired. Davies
and Hunter [7] derived an equation to correct for inertia which has been
shown to be fairly accurate. It is recommended this equation be incor-
porated in the existing data reduction algorithm. A Fortran subroutine has
been written by the author, is included in Appendix E, and can be directly
integrated in the existing data reduction programs in use at the University of
Florida and at Tyndall AFB.


By following the guidelines in steps 1 through 4, the test engineer has
in effect removed or corrected for any rate, friction, or two-dimensional
effects due to the testing apparatus itself, and thus the results will permit the




81


observation of the true material behavior. The intrinsic characteristics of
the material will be separated from any other effects inherent to the way the
specimen is being tested. Therefore, the Split Hopkinson Pressure Bar has
been and remains a reliable, accurate and useful tool to study material
response to loading, even for a new proposed system 6 inches in diameter.













REFERENCES


1. Kolsky, H., "An Investigation of the Mechanical Properties of Materials
at Very High Rates of Loading," Proceedings of the Physics Society, Vol.
62, pp. 676-700, 1949.


2. Hauser, F.E., Simmons, J.A., and Dorn, J.E., "Response of Metals to
High Velocity Deformation," Proceedings of the Metallurgical Society
Conference, Vol. 9, pp. 93-101, 1960.


3. Lindholm, U.S., "Some Experiments with the Split Hopkinson Pressure
Bar," Journal of the Mechanics and Physics of Solids, Vol. 12, pp.
317-335, 1964.


4. Malvern, L.E., and Ross C.A., "Dynamic Response of Concrete and
Concrete Structures," Final Technical Report, AFOSR Contract
F49620-83-K007, University of Florida, Gainesville, FL, May 1986.


5. Robertson, K.D., Chou, S. and Rainey, J.H., "Design and Operating
Characteristics of a Split Hopkinson Pressure Bar Apparatus," Techni-
cal Report AMMRC TR 71-49, Army Materials and Mechanics Re-
search Center, Watertown, MA, November 1971.


6. Kolsky, H., Stress Wave in Solids, Dover Publications, New York, 1963.


7. Davies, E.D.H., and Hunter, S.C., "The Dynamic Compression Testing of
Solids by the Method of the Split Hopkinson Pressure Bar," Journal of
the Mechanics and Physics of Solids, Vol. 11, pp. 155-179, 1963.


8. Rand, J.L., "An Analysis of the Split-Hopkinson Pressure Bar," U.S.
Naval Ordnance Laboratory, White Oak, MD, NOLTR67-156, 1967.


9. Dharan, C.K.H. and Hauser, F.E., "Determination of Stress-Strain Char-
acteristics at Very High Strain Rates," Experimental Mechanics, Vol. 10,
pp. 370-382, 1970.







10. Samanta, S.K., "Dynamic Deformation of Aluminum and Copper at
Elevated Temperatures," Journal of the Mechanics and Physics of
Solids. Vol. 19, pp. 117-124, 1971.


11. Jahsman, W.E., "Re-examination of the Kolsky Technique for Measuring
Dynamic Material Behavior," Journal of Applied Mechanics, Vol. 38, pp.
75-82, 1971.


12. Chiu, S.S. and Neubert, V.H., "Difference Method for Wave Analysis of
the Split Hopkinson Pressure Bar with a Viscoelastic Specimen", Jour-
nal of the Mechanics and Physics of Solids. Vol. 15, pp. 177-193, 1967.


13. Young, C. and Powell, C.N., "Lateral Inertia Effects on Rock Failure in
Split Hopkinson-Bar Experiments," 20th U.S. Symposium on Rock
Mechanics, Austin, TX, pp. 299-303, 1979.


14. Nicholas, T., "An Analysis of the Split Hopkinson Bar Technique for
Strain-Rate-Dependent Material Behavior", Journal of Applied
Mechanics. Vol. 1, pp. 277-282, March 1973.


15. Lawson, J.E., "An Investigation of the Mechanical Behavior of Metals at
High Strain Rates in Torsion," Ph.D. Dissertation, Air Force Institute of
Technology, Wright-Patterson AFB, OH, June 1971.


16. Baker, W.E. and Yew, C.H., "Strain-Rate Effects in the Propagation of
Torsional Plastic Waves", Journal of Applied Mechanics, Vol. 33, pp.
917-923, 1966.


17. Jones, R.P.N., "The Generation of Torsional Stress Waves in a Circular
Cylinder", Quarterly Journal of Mechanics and Applied Mathematics,
Vol. 12, pp. 208-211, 1959.


18. Okawa, K., "Mechanical Behavior of Metals Under Tension-Compres-
sion Loading at High Strain Rate," International Journal of Plasticity,
Vol. 1, pp. 347-358, 1985.


19. Ross, C.A., Nash, P.T., Friesenhahn, G.H., "Pressure Waves in Soils
Using a Split-Hopkinson Pressure Bar," Final Technical Report, ESL-
TR-81-29, AFESC, Tyndall AFB, FL, July 1986.








20. Rajendran, A.M., and Bless, S.J., "High Strain Rate Material Behavior,"
Technical Report, AFWAL-TR-85-4009, AFWAL, Kirtland AFB, NM,
December 1985.


21. Ross, C.A., "Split Hopkinson Pressure Bar Tests," Final Technical
Report, ESL-TR-88-82, AFESC, Tyndall AFB, FL, March 1989.


22. Follansbee, P.S. and Frantz, C., "Wave Propagation in the Split Hopkin-
son Pressure Bar", ASME Journal of Engineering Materials and Tech-
nology, Vol. 105, pp. 61-66, 1983.


23. Redwood, M., Mechanical Waveguides, Pergamon Press, New York, 1960.


24. Bertholf, L.D. and Karnes, C.H., "Two-dimensional Analysis of the Split
Hopkinson Pressure Bar System," Journal of the Mechanics and Physics
of Solids, Vol. 23,pp. 1-19, 1975.


25. Bancroft D., "The Velocity of Longitudinal Waves in Cylindrical Bars,"
Physical Review, Vol. 59, pp. 588-593, April 1941.


26. Mindlin, R.D. and Herrmann G., A One-Dimensional Theory of Com-
pressional Waves in an Elastic Rod, Pro. 1st U.S. Nat. Cong. App.
Mech., Chicago, pp. 187-191, 1951.


27. Davies, R.M., "A Critical Study of the Hopkinson Pressure Bar,"
Philosophical Transactions Royal Society, Vol. A240, pp. 375-457, 1948.


28. Kolsky, H., "The Propagation of Stress Pulses in Viscoelastic Solids,"
Phil. Mag. Vol. 1, pp. 693-710, 1956.


29. Love, A. E. H., 1927, The Mathematical Theory of Elasticity, Dover
Publications, New York.


30. Karal, F.C., "Propagation of Elastic Waves in a Semi-infinite Cylindrical
Rod Using Finite Difference Methods," Journal of Sound Vibration,
Vol. 13, pp. 115-145, 1970.








31. Gong, J.C., Malvern, L.E., and Jenkins, D.A., "Dispersion Investigation
in the Split Hopkinson Pressure Bar," Journal of Engineering Materials
and Technol ogy, Vol. 112, pp. 309-314, July 1990.


32. Jerome, E.L., "Feasibility of a 6-inch Split Hopkinson Pressure Bar
(SHPB)," Final Technical Report, ESL-TR-90-44, AFESC, Tyndall
AFB, FL, August 1990.


33. Hopkinson, B., "A Method of Measuring the Pressure Produced in the
Detonation of High Explosives or by the Impact of Bullets," Philosophi-
cal Transactions of the Royal Society of London, Vol. 213, pp. 437-456,
1914.


34. Osborne, J.J., and Matuska, D.A., "HULL, Finite Difference Code,"
Orlando Technology, Inc., Shalimar, Florida, 1987.


35. Rinehart, J.S., Stress Transients in Solids, HyperDynamics, Santa Fe,
New Mexico, May, 1975.


36. Chree, C., "The Equations of an Isotropic Elastic Solid in Polar and
Cylindrical Coordinates, Their Solutions and Applications," Cambridge
Philosophical Society, Transactions, Vol. 14, 1889, pp.250-369.


37. Pochhammer, L., "On the Propagation Velocities of Small Oscillations
in an Unlimited Isotropic Circular Cylinder," Journal fur die Reine und
Angewandte Mathematik, Vol. 81, 1876, pp. 324-326.













APPENDIX A
MODIFIED POCHHAMMER-CHREE PROGRAM AND SAMPLE OUTPUTS

This program uses the Pochhammer-Chree frequency equation solu-
tion to the two-dimensional equations of motion in cylindrical coordinates
to analyze the wave propagation of an arbitrary trapezoidal input pulse in a
bar of varying dimensions and material properties. The wave is carried
through a typical compressive SHPB. Multiple reflections and transmissions
at the interfaces between the specimen and bars are calculated and the
waves reconstructed in the incident and in the transmitter bar.
The program logic is organized as follows:
The user is asked to enter the areas, densities, and wave speeds for
the bars and the specimen. This information is used to calculate the charac-
teristic impedances of the materials.
The user is asked to enter the pulse duration, number of time steps
(each 0.5e-6 seconds long), the diameter of the bar, and the dimensions of
the specimen.
The input pulse is represented via a Fourier series. 55 terms are
retained. Amplitude of the incident pulse is normalized to 1.0. Each com-
ponent has a unique phase angle, frequency, and amplitude and they are all
in phase at the onset.
The Pochhammer-Chree solution for mode M1,i is used to calculate

the wave speed of each Fourier component in the bar material as well as in
the specimen material.
Each pulse is then started at a distance of 2 meters from the
specimen and is followed to the incident bar-specimen interface through the
multiple reflections and transmissions that occur at that point and is fol-
lowed into the specimen to the specimen-transmitter bar interface where
more reflections and transmissions take place. Since the speed of each
86







component of the pulse is known, and since the characteristic impedances
are known, the program calculates the amount that gets reflected and trans-
mitted each time, as well as how long it takes for each component to travel
any distance. All 55 terms are treated in this way, and then the waveform is
reconstructed both in the incident and in the transmitter bar at the strain
gage locations both 1 meter from the specimen.
The output is in numerical form. The incident, transmitted, and
reflected waves are represented by values one-half microsecond apart.
These data can be easily plotted with any available software. LOTUS 123
was used by the author.
Additionally, the program calculates the cross sectional variations of
the stresses derived from the Pochhammer-Chree frequency solution. Be-
ssel functions are evaluated numerically. For each radial value the 55 terms
making up the traveling wave are added for each time increment. The por-
tion of each stress dependent on r, discussed in chapter 3, are essentially
multiplied to each Fourier component at each time step. All the terms, at
their particular frequency are added up in time to form the complete
waveform at each radial location. Longitudinal, radial, and shear stresses
are calculated, and their amplitudes are all normalized to the longitudinal
stress at the center.
Figures 36 and 37 are examples of cross sectional variations of the
longitudinal stress for a 2-inch and a 6-inch SHPB. Reflected and trans-
mitted pulses are shown.





88


















It)























0 0
v)





oo N






0 0 0 N tla 0 0 u


< 8 o
7)0 a a 7 o N 5 -
















S 0c
/ C)
Mo. r




V) m
-JJ
1 I T- o







Lu I (n ^ I P S S c






0



l C3 0

d a 0 0 0 0i0








































































Sc3WUS O3iKSNVil


Si0
? ?

UU)



CC

















F
F1
S 5 S O21Sn5 T 5 L




















LL)
-J












ss 03 ,. a
0al CG'lu 0- -S
0-- "OI
3- ^^-- s 1
Z i \ =s 1
. .^
i ^ \ 1 i
na\ *^ ES V ,3 i ||









C THIS PROGRAM USES POCHHAMMER AND CHREE
C FREQUENCY SOLUTION OF THE TWO DIMENSIONAL
C EQUATIONS OF MOTION IN CYLINDRICAL COORDINATES
C TO ANALYZE THE WAVE PROPAGATION OF AN ARBITRARY
C TRAPEZOIDAL INPUT PULSE IN A BAR OF VARYING
C DIMENSIONS AND MATERIAL PROPERTIES.
C
C FURTHERMORE, IT CALCULATES THE RESPONSE DUE
C TO THE MULTIPLE REFLECTIONS AND TRANSMISSIONS
C AS THE WAVE ENCOUNTERS A SPECIMEN OF DIFFERENT
C MATERIAL, SAME DIAMETER, AND SPECIFIED LENGTH.
C
C THE OUTPUT CONSISTS OF THE INCIDENT, REFLECTED
C AND TRANSMITTED PULSES AT STATIONS ONE METER
C FROM THE SPECIMEN BOTH ON THE INCIDENT AND ON THE
C TRANSMITTER BAR. SOLUTIONS ARE GIVEN AT THE CENTER
C OF THE BAR (FILE FOR001.DAT), AND AT THE SURFACE
C (FILE FOR010.DAT) TO CHECK FOR RADIAL UNIFORMITY
C OF THE MOTION. ALL AMPLITUDES ARE NORMALIZED.
C
dimension d(3000,40),f(3000),phl(200),ph2(200)
dimension time(3000,40),u(3000),fr(3000),c(200),dl(200)
dimension frr(3000),ut(3000),frrr(3000),utt(3000)
dimension sig(100,200)
s-0.014
pi-3.14159
C WAVE SPEEDS, AREAS, AND DENSITIES ARE ENTERED TO
C CALCULATE CHARACTERISTIC IMPEDANCES

write(*,*)' input speeds, densities, and areas'
write(*,*)' col,co2,(m/sec),rol,ro2,A1,A2'
write(*,*)' bar is 1, specimen is 2'
Read(*,* )col,co2,rol, ro2,al,a2

a-2.0/(pi**2*s)

tr-2*al*ro2*co2/(ro2*co2*a2+rol*col*al)
ref-(a2*ro2*co2-al*rol*col)/( ro2*co2*a2+rol*col*al)
tr2=2*a2*rol*col/( ro2*co2*a2+rol*col*al)
ref2-ref

write(*,*)tr,ref,tr2,ref2
n=-1
write(*,*)' input nn,wo (1/sec),diameter (mm),spec. length(m)'
read(*,*)nn,wo,diam,zl2
dia-diam/1000.0
C STRAIN GAGE LOCATION IS ASSUMED AT ONE METER
C FROM IMPACTED END, AND ONE METER FROM EITHER
C SIDES OF THE SPECIMEN
zl-1.0
213-1.0
z24-1.0

C 55 TERMS ARE USED IN THE FOURIER ANALYSIS

imm-55
do 5 j-l,mm
n=n+2
ne-(n-1)/2










C CALCULATE THE AMPLITUDES OF EACH PULSE
C AS THEY CROSS THE INTERFACES BACK AND FORTH
C
C AT STATION ON THE INCIDENT BAR:

d(j,1)=(-a*(-1)**ne/n**2)*sin(n*pi*s)
d(j,2)=ref*d(j,1)
c THIS NEXT PIECE IS THE FIRST REFLECTION BACK
C WHICH WILL ADD TO THE REFLECTED STRESS
d(j,3)=tr*ref2*tr2*d(j,l)
d(j,6)=d(j,3)*ref2**2
C AT STATION ON THE TRANSMITTER BAR:

d(j,4)=tr*tr2*d(j,1)
c PLUS WHAT BOUNCES BACK AND FORTH IN THE SPECIMEN
C AND ADDS TO THE TRANSMITTED STRESS
d(j,5)=tr*ref2*ref2*tr2*d(j,1)
d(j,7)=d(j,5)*ref2**2

5 continue

C SUBROUTINE ANGLE CALCULATES PHASE CHANGES, VELOCITIES
C AND WAVELENGTHS FOR EACH TERM IN THE FOURIER SERIES

call angle(dia,col,co2,wo,nim,phl,ph2,c,dl)
C SUBROUTINE CROSS CALCULATES THE CROSS SECTIONAL VARIATION
C OF THE DISPLACEMENTS AND STRESSES. ONLY LONGITUDINAL STRESSES
C ARE BROUGHT BACK INTO THE MAIN PROGRAM
C SIGMA RR AND SIGMA ZZ ARE OUTPUTED TO FILE FOR002.DAT

call cross(dia,c,dl,mm,sig)
c
c CALCULATE THE TIMES EACH PULSE TAKES TO
C REACH THE STRAIN GAGES

n-i
do 10 j-l,mm
n-n+2
time(n,l)=zl*((-1.0/col)+phl(n))
time(n,2)=time(n,l)+2.0*z13*((-1.0/col)+phl(n))
time(n,3)=time(n,2)+z12*( (-1.0/co2)+ph2(n))*2.0
time(n,4)=time(n,l)+z24*2.0*( (-.0/col)+phl(n))
1 +zl2*((-1.0/co2)+ph2(n))
time(n,5)=time(n,4)+2*z12*((-1.0/co2)+ph2(n))
time(n,6)=2*z12*ph2(n)/(0.5*1.e-6)
10 continue




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