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This is to certify that the

thesis entitled

CONSTRUCTION, CALIBRATION, AND
APPLICATION OF A SPLIT HOPKINSON PRESSURE BAR

presented by
Guojing Li

has been accepted towards fulﬁllment
of the requirements for

MS degree in MechanicalJEng ineering

 

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Major professor

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TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJClRC/DatODquBS-DJS

CONSTRUCTION, CALIBRATION AND APPLICATION OF
A SPLIT HOPKINSON PRESSURE BAR

By

Guojing Li

A THESIS
Submitted to
Michigan State University

In partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2002

ABSTRACT

CONSTRUCTION. CALIBRATION AND APPLICATION OF
A SPLIT HOPKINSON PRESSURE BAR

By

Guojing Li

The behavior of materials at high strain rates is different from that under
quasi-static loading. Among the experimental techniques for dynamic behavior of
materials, Split Hopkinson Pressure Bar (SHPB) is the most common technique
for dynamic stress-strain characterizations due to its capability of producing a
large range of nearly uniform strain rates. Although many SHPBs have been
constructed, there is no step-by-step guidelines for SHPB construction, neither is
there a standard design. It is the primary goal of this research to construct an
SHPB for characterizations of materials, especially thick laminated composites.
Thick laminated composites have different material properties from thin
counterparts because the thermal cycle for curing polymer matrix is not
necessarily uniform through the thickness of thick laminated composites.
Consequently, when subjected to dynamic loading, thick laminated composites
behave differently from thin counterparts. Experimental results based on SHPB
have veriﬁed the difference of the properties close to the surface and those close
to the midplane of laminated composites. In addition, it has been found from this
research that stress-strain curves are strongly affected by the specimen

dimensions.

ACKNOWLEDGEMENTS

This thesis represents a culmination of hundreds of hours work by its
author, Dr. Dahsin Liu from Michigan State University and many previous
investigators. I would like to thank Dr. Liu for his insight and directions. Dr. Liu
directed my thesis research step by step from the very beginning and helped me
to rewrite the thesis. I would like to thank Dr. Gary Cloud and Mr. Brian Wright for
the directions on the circuit designs. I would like to thank the tutors in the writing
center of Michigan State University for taking time to review my thesis. I would
also like to thank Sen Li, my wife, and Xinui Li, my daughter, for all of their love
and understanding, though they probably don’t know it. Thanks to all of you, I

appreciate everything you have done for me.

iii

TABLE OF CONTENTS

LIST OF TABLES ................................. vii
LIST OF FIGURES ........................................................................... viii
CHAPTER 1
INTRODUCTION ................................................................................ 1
1. Background ............................................................................. 1
2. Statement of the Problem and Objectives ....................................... 3
3. Organization of the Thesis ........................................................... 4
CHAPTER 2
CONSTRUCTION OF A SPLIT HOPKINSON PRESSURE BAR ................... 6
1. The Pressure Source System ....................................................... 6
2. The Split Bar System .................................................................. 8
3. The Data Acquisition System ................................. ' ..................... 10
3.1 Wheatstone Bridge Circuit .................................................... 10 I
3.2 Differential Ampliﬁer ............................................................ 12
3.3 Computer-Aided Data Processing .......................................... 12
4. Overall Operation of the SHPB .................................................... 14
5. Detailed Operation Procedures .................................................... 16
5.1 Starting Up Procedures ....................................................... 16
5.2 Shutting Down Procedures ................................................... 18
CHAPTER 3
CALIBRAIONS OF COMPONENTS AND SHPB ....................................... 19
1. Calibration of Differential Ampliﬁers .............................................. 19

iv

2. Calibration of Wheatstone Bridge Circuits (with Differential Ampliﬁer)....22

3. Calibration of the Whole SHPB ...................................................... 25
3.1 Calibration with Load Cell ................... _ .................................... 25
3.2 Calibration with Instrumented Striker Bar ................................... 28
3.3 Calibration by Testing 6061-T6 Aluminum .................................. 34
CHAPTER 4
CHARACTERIZATIONS OF THICK LAMINATED COMPOSITES .................. 46
1. Specimen Preparation .................................................................. 46
2. Experimental Results ................................................................... 49
2.1 Strain-Rate Effect .................................................................. 49
2.2 Laminated and Assembled Specimens ...................................... 49
2.3 Effect due to LID Ratio ........................................................... 53
2.4 Effect due to Specimen Inhomogeneity ..................................... 53
2.5 Young’s Modulus and Yielding Point ..... ........................ 54
CHAPTER 5
DISCUSSIONS .................................................................................. 61
1. Dimensions of Bars ..................................................................... 61
1.1 Bar Diameter Dimr ................................................................... 61
1.2 Bar Length Lba, ...................................................................... 62
2. Dimensions of Specimens ............................................................. 63
2.1 Specimen Diameter D ............................................................. 63
2.2 Specimen Length L ................................................................ 63
3. Constant Strain Rate ................................................................... 64

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS .......................................... 67
1. Conclusions .............................................................................. 67
2. Recommendations .................................. I ................................... 68
APPENDICES ......................................................................... ‘ ........... 7 0
APPENDIX A
OTHER FIGURES AND WAVE SIGNALS ....................................... 71
APPENDIX B
SPLIT HOPKINSON PRESSURE BAR THEORY ............................ 104
REFERENCES .................................................................................. 111

vi

LIST OF TABLES

Table 1 Density, wave speed and Young’s modulus of split bars ................. 31
Table 2 Dimensions and dimensional ratios of aluminum specimens ........... 42
Table 3 Comparison of Yielding Points .................................................. 45

Table 4 Dimensions and dimensional ratios of Glass/Epoxy specimens ....... 47

Table 5 Parameters of SHPB and U0 of specimen used by
various researchers ................................................................ 65

vii

LIST OF FIGURES

Figure 1 Schematic diagram of a split Hopkinson pressure bar ..................... 7

Figure 2 Teﬂon bearing and associated frame and rail ................................. 9
Figure 3 Detailed design of a Wheatstone bridge circuit .............................. 11
Figure 4 Detailed design of a differential ampliﬁer ...................................... 13

Figure 5 Overall operation and data acquisition and
processing of the split Hopkinson pressure bar .............................. 15

Figure 6 Output wave (below) of an initially sqaure wave (top) with

a frequency of 3500 Hz from a differential ampliﬁer ........................ 20
Figure 7 Frequency response function of ampliﬁer ...................................... 21
Figure 8 Gain of ampliﬁer based on square waves at 3500 Hz ..................... 21
Figure 9 Wheatstone bridge circuit and shunt calibration resistors ................. 23

Figure 10 (a), (b) Comparisons between experimental
measurements and theoretical calculations ......... ‘ ........................ 24

Figure 11 Schematic diagram of striker bar impacting on load cell ................. 27

Figure 12 Comparison between the signals from load cell
and those from strain gages in striker bar .................................... 29

Figure 13 (a-c) Schematic diagram of the striker bar
impacting the incident bar, the transmitter bar
and the combination of the two bars .......................................... 30

Figure 14 The discrepancies of strain measurements
between the incident bar and the striker bar and that
between the transmitter bar and the striker bar ........................... 32

Figure 15 Wave speed as a function of gas pressure.
(a) Striker bar impacting the combination of incident
and transmitter bars.
(b) Striker bar impacting the incident bar.
(c) Striker bar impacting the transmitter bars ............................... 33

viii

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Figure 22

Figure 23

Figure 24

Figure 25

Figure 26

Figure 27

Figure 28

Figure 29

Figure 30

(a), (b) Strain waves showed on oscilloscope during 500 ms

(c), ((1) Time durations for the strain waves

with 50% of amplitude reduction ............................................ 35
Typical incidence, reﬂection and transmiSsion waves from

SHPB for Aluminum 6061-T6 ................................................ 36
Schematic diagram of time-positon relation of the strain waves

in the incident bar, specimen and transmitter bar ...................... 37
Input and output force histories of four aluminum

specimens with the same impact gas pressure .......................... 41
The stress-strain curves of aluminum 6061-T6

under different strain rates ................................................... 44
Bi-metal hole saw ............................................................... 48
Typical incidence, reﬂection and transmission waves from

SHPB for glass-epoxy composite .......................................... 50
Stress-strain curves for glass-epoxy composites

with similar dimensions and U0 ratios (1.6 ) ............................ 51

Stress-strain curves for two groups of glass-epoxy composites
with similar strain rates and total dimensions.

Group one—from thick glass-epoxy composite plates,

Group two—from thin glass-epoxy composite plates

and assembled .................................................................. 52
Stress-strain curves for glass-epoxy composites

with similar strain rates but different UD ratios .......................... 55
Stress-strain curves for glass-epoxy composites

with different strain rates and different UD ratios ...................... 56
Stress-strain curves for glass-epoxy composites

with similar strain rates but different UD ratios ......................... 57
Stress-strain curves for three groups of glass-epoxy

composites with similar dimensions and UD ratio (0.8) .............. 58
Young's Modulus vs. strain rates at different UD ratios ............... 60
Yielding points vs. strain rates at different L/D ratios ................... 60

ix

Figure 31 Stress-strain curves at constant strain rates for the
composte specimens with D=15.8 mm, L=25.4 mm.

(based on various levels of constant incident strain wave tests ....... 66
APPENDICES
Figure A-1 Other stress-strain curves of aluminum 6061-T6
with different strain rates ...................................................... 71
Figure A-2 Stress-strain curves of glass-epoxy composite
specimens cutting from the same plate .................................... 72
Figure A-3 Stress-strain curves of assembled glass-epoxy
composite with different layers ............................................. 73
Figure A-4 Stress-strain curves of glass-epoxy
composite with different UD ratios ......................................... 74
Figure A-5 Calculation of wave speed in bars ......................................... 75
Figure A-6 Wave shapes and forces comparison of Aluminum
6061-Y6 with shaping technique. (from Dr. W. Chen [5]) ............ 76
Figure A-7 to Figure A-12
Incident, reﬂected and transmitted waves for the aluminum
6061-T6 specimen with various diameters and lengths ............... 77
Figure A-13 to Figure A-33
Incident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with various diameters and lengths ........... 83
Figure A-34 Pressure bar shown with differential element ........................ 104
Figure A-35 Parameters of cylindrical specimen and bars ........................ 107

CHAPTER 1

INTRODUCTION

1. Background

The behavior of materials at high strain rates is different from that under
quasi-static loading. As many materials are used for high-performance stuctures
and are subjected to dynamic loading, the dynamic behavior of the materials has
become a primary concern in structural designs. A few experimental techniques
have been developed for characterizations of dynamic behavior of materials, e.g.
drop-weight tower, plate impact, split Hopkinson pressure bar (SHPB) and high-
speed photography.

Based on a force transducer, the drop-weight tower is a relatively simple
testing technique. However, the strain rate produced by the technique of drop-
weight tower is limited by the dropping height of the weight. In contrast, the plate
impact technique is able to produce strain rates as high as 106 s‘1 and 107 s“. It
uses very thin specimens, such as coating a thin ﬁlm of specimen onto a plate,
for impact. Different from direct impacts on specimens like those occur in the
drop-weight tower and plate impact techniques, SHPB uses two bars to convey
impulsive waves into specimens and to measure the wave signals input into
(incident) and output from (reﬂected and transmitted) the specimens. SHPB is
useful for testing specimens at intermediate strain rates, i.e. between 100 s'1
and 104 3". Almost all experiments utilizing the drop-weight tower, plate impact

and SHPB use electrical-based devices for signal sensoring. With the

advancements in high-speed cameras and optical methods, such as holographic
interferometry and cautics, high-speed photography has also been developed for
characterization of dynamic behavior of materials. .

Among the four techniques mentioned above, SHPB is the most common
technique for dynamic stress—strain characterizations due to its capability of
producing a large range of nearly uniform strain rates. The Pressure Bar
technique was initiated by Hopkinson in the early 1900’s [1]. He used a long rod
to convey a force pulse to a force transducer. However, it was not until 1949 that
Split Pressure Bar was proposed by Kolsky. He used two Hopkinson’s pressure
bars to measure the dynamic signals of materials in compression. From then on,
Split Hopkinson Pressure Bar has become a standard technique for determining
dynamic properties of materials, verifying the constitutive models of materials,
identifying the propagation of plastic waves in materials, etc. Since 1949, the
SHPB technique has advanced in many aspects. However, the following
fundamental assumptions remain [2]. They should be carefully observed to
obtain meaningful results from SHPB tests.

(1) The split bars should remain within elastic range during operations.

(2) The waves propagated in the bars should be one-dimensional longitudinal
type-

(3) The deformation in the specimen should be uniform through the length of the
specimen.

During the past ﬁve decades, the advancements of the SHPB technique

include the development of instrumentations to detect, process and display the

signals with minimum distortion, the reﬁnement of basic SHPB theories and the

extensions of SHPB in characterizing various material properties, such as

tensional, torsional, shear and fracture properties, at high strain rates and as

effected by temperature . Some issues concerning the basic SHPB theories have

been revisited again and again by many researchers and are worth extra

attention in the construction and application of SHPB [1-3], e.g.

(1) the uniformity of stress in the specimen,

(2) the effect of strain-rate history on the the microstructure of the specimen

material,

(3) the effect of D/Dba, ratio (D is the diameter of the specimen) on the
assumption of one-dimensional wave propagation,

(4) the effect of wave dispersion and distortion on the stress-strain curve,

(5) the effects of friction and radial inertia on the one-dimensional aSsumption.
Although SHPB has become a standard testing technique and many SHPBs

have been constructed over the years, there is neither step-by-step guideline for

SHPB construction nor a standard design. As the electronic instrumentation has

experienced rapid advancement in recent years, the data acquisition, processing

and presentation have become easier, more efﬁcient and more accurate. It is the

primary goal of this thesis research to construct a SHPB for. material

characterization.

2. Statement of the Problem and Objectives of the Thesis

Owing to their high stiffness and high strength with low density, ﬁber-
reinforced polymer matrix composite materials are excellent candidate materials
for high-performance structures. With increasing number of applications using
composite materials, more and more thick composite plates are used for
structural applications. However, thick composite plates may have different
material properties from thin counterparts as the thermal cycle for curing polymer
matrix is not necessarily uniform throught the thickenss of thick composite plates.
Consequently, when subjected to dynamic loading, the thick composite plates
may behave differently from the thin counterparts. The primary goal of this thesis
research is to characterize the compressive behavior of thick laminated
composites at high strain rates. In order to achieve the goal, the following efforts
are identiﬁed:

(1) to construct a Split Hopkinson Pressure Bar,

(2) to calibrate the Split Hopkinson Pressure Bar,

(3) to use the Split Hopkinson Pressure Bar to characterize thick laminated
composites,

(4) to analyze the experimental results.

3. Organization of the Thesis

In addtion to the introductory chapter, Chapter 1, the thesis is divided into ﬁve
chapters. Chapter 2 describes the construction of a Split Hopkinson Pressure
Bar. Overall setting-up procedures and operation procedures are also given.

Chapter 3 presents the calibration procedures in validating the Split Hopkinson

Pressure Bar. Calibrations for both individual components and the whole SHPB
are included. Chapter 4 presents the application of the SHPB in the
characterization of thick laminated composites. Specimens with different
dimensions and dimensional ratios are investigated. Chapter 5 presents
discussions on the effects of dimensions and dimensional ratios of bars and
specimens. The conclusions from the thesis research are summarized in Chapter

6. Some recommendations for future research are also identiﬁed.

CHAPTER 2

CONSTRUCTION OF A SPLIT HOPKINSON PRESSURE BAR

Split Hopkinson Pressure Bars (SHPBs) have been commonly used for
characterizations of material behavior at high strain rates. An SHPB usually
consists of three major component systems: a pressure source system, split bar
system and data acquisition system. The schematic diagram of the Split
Hopkinson Pressure Bar constructed in the thesis research is given in Figure 1.

The three component systems can be clearly identiﬁed in the diagram.

1. The Pressure Source System

The pressure source system provides required pressure for accelerating the
striker bar. A gas source system, originally used for a gas gun, was used as the
pressure source system in the Split Hopkinson Pressure Bar. It includes a
cylindrical tank for storing nitrogen gas, a small gas chamber for storing the
required amount of gas in each operation, a piping-and-valve system for
controlling the pressure of the operating gas and an electro-magnetic valve for
controlling the gas releasing. The gun barrel of the original gas gun was retained
for gas releasing.

In the piping-and-valve system, there are two pressure gauges; one is used
for measuring gas pressures up to 3000 psi (21 MPa) and the other is used for
measuring gas pressures lower than 100 psi (700 kPa). The former is for large-

scale adjustment while the latter ﬁne adjustment. Due to the strength of the

 

 

 

 

 

 

 

 

 

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Figure 1 Schematic diagram of a split Hopkinson pressure bar.

pipes, the maximum operating pressure of the piping-and-valve system is set at

400 psi (2.8 MPa).

2. The Split Bar System

The split bar system is the loading and measuring device used for
characterizations of specimens at high strain rates. It includes a striker bar, an
incident bar and a transmitter bar. All three bars are made of 347 Stainless Steel
with high stiffness 27.6 Msi (193 GPa), high strength 29.3 ksi (205 MPa) and
have an identical diameter Dba, of 0.75" (0.019 m). The lengths of the incident bar
and the transmitter bar Lb,r are 33" (0.838 m), resulting in an Lbar/DbaF 44,
whereas that of the striker bar Lsmker is 7" (0.178 m), resulting in an LbarlLsmkeF
4.7.

In order to hold the incident bar and the transmitter bar horizontally with
minimum friction and deﬂection and to achieve excellent alignment of the bars,
two Teﬂon bearings were built for each bar. The Teﬂon bearings were framed
and ﬁxed on an aluminum rail, which was joined to a solid steel foundation.
Details of the Teﬂon bearing and associated frame and rail are depicted in Figure
2.

Since the gun barrel (for gas releasing) has a diameter of 0.5" (12.5 mm),
which is different from the diameter of the striker bar (0.75"), a joining component
was built and attached to the end of the gun barrel to accommodate the striker
bar. A small clearance around 0.04" (1 mm) was maintained between the striker

bar and the joining component. A gap of 1" (0.025 m) was also kept between the

 

 

 

    

 

 

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Figure 2 Teﬂon bearing and associated frame and rail

striker bar and the incident bar. Both the clearance and the gap were required to
warrant a complete separation of the striker bar from the joining component
during the impact of the striker bar onto the incident bar. In addition, a relatively
soft material was installed at the very end of the transmitter bar as a stopper to

protect the transmitter bar from overshooting.

3. The Data Acquisition System

The functions of the data acquisition system are to detect, process and
present the strain signals in the incident bar and the transmitter bar. The system
consists of a Wheatstone bridge circuit and an ampliﬁer for each bar. The signals
output from the ampliﬁer are then input into a computer-based oscilloscope for

data processing and display.

3.1 Wheatstone Bridge Circuit

Electrical-resistance strain gages EA-06-062AQ-350 manufactured by
Measurements Group were used for strain wave measurements. Two strain
gages were mounted on the mid-span of each bar at opposite sides to measure
the strain waves. They were called active gages. Two strain gages of the same
type, so-called dummy gages, were added to form a four-arm Wheatstone bridge
circuit. Figure 3 shows the details of a Wheatstone bridge circuit. An adjustable
resistor ranging from 0 to 25 Q was used for initial circuit balancing. A resistor of
510 Q was also inserted in the circuit for circuit stability. Since the two active

gages are located on the opposite side of each bar and on the opposite arm of

10

 

 
   

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11

each circuit, signals caused by bar bending can be automatically eliminated in
the signal processing. Besides, the use of two active gauges has sensitivity twice

that of one active gauge.

3.2 Differential Ampliﬁer

Since the signals from the Wheatstone bridge circuits are usually very small,
a differential ampliﬁer was designed to magnify the signals output from each
Wheatstone bridge circuit. Figure 4 shows the design details of a differential
ampliﬁer. A TL084CN ampliﬁer chip (outlined by the dotted lines) is the core of
the design. A resistor of 47 kQand a resistor of 1 MQwere used to produce a
magniﬁcation factor around 20 (1 MQ/47 k9 ), which was determined based on
the signal output from the Wheatstone bridge circuit and required by the digital
oscilloscope. Besides, a DC power supply capable of providing :15 V was used
due to the possibility of having positive and negative strains during experimental

measurements.

3.3 Computer-Aided Data Processing

Triggering technique is an important issue in dynamic measurements. Since
an electrical disturbance with a voltage around 100 mV was usually generated
from switching the electro-magnetic valve used, the electrical disturbance was
used as a triggering source to initiate the recording of the wave signals. In signal
acquisition, a commercial circuit product 5112 Digital Oscilloscope manufactured

by National Instrument was used. The Digital Oscilloscope has two channels and

12

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Figure 4 Detailed design of a differential amplifier.

13

+15V DC

III;

a total sampling rate of 100 million samples per second. With a measuring
duration of 0.5 ms, each channel can acquire as many as 25,000 samples, which
is sufﬁcient for subsequent data manipulation. Forexample, Microsoft Excel was
used for data integration. Once the signals were processed, they were displayed
on a computer monitor. The digitized data was also stored in the computer for

later applications

4. Overall Operation of the SHPB

Figure 5 gives a ﬂow chart of overall operation procedures for the split
Hopkinson pressure bar. Initially, all mechanical components should be adjusted
and electronic components should be engaged, e.g. aligning the split bars,
installing a specimen between the incident bar and the transmitter bar, adjusting
the needed gas pressure, setting up the striker bar, and turning on the computer
and power supplies for the Wheatstone bridge circuits and the ampliﬁers. In
setting up the SHPB, the alignment and leveling of the individual bars were
perhaps the most important step. In this procedure, a laser beam was used for
the alignment while a level rule was used to adjust the level of the bars. In
specimen installation, both specimen ends were carefully polished to ensure that
they were parallel to each other. A lubricant was applied to both surfaces to
ensure low friction (caused by the transverse enlargement due to Poisson’s
effect) and close contact between the specimen and the bars, hence the one-

dimensional wave propagation through the aligned bars and specimen.

l4

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record signals corresponding to
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11

 

record Signals corresponding
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Figure 5 Overall operation and data acquisition and processing of
the split Hopkinson pressure bar.

15

Once the impact between the striker bar and the incident bar takes place, a
strain wave will be generated and propagate through the strain gages mounted
on the incident bar. A part of the wave will subsequently pass through the
specimen and reach the strain gages mounted on the transmitter bar while
another part of the wave will be reﬂected from the front end of the specimen and
again be recorded by the strain gages mounted on the incident bar. The strain
waves detected by the strain gages will be ampliﬁed and input to the computer
for data processing. Since the strain wave recorded by the strain gages mounted
on the incident bar is associated with the strain history and since that recorded
by the strain gages mounted on the transmitter bar is associated with the stress
history, a stress-strain relation can be established for the specimen under the
investigation of the strain rate, which is also associated with the strain wave

recorded by the strain gages mounted on the incident bar.

5. Detailed Operating Procedures
5.1 Starting Up Procedures
A. Computer-based Oscilloscope

a. Turn on the computer. Run the NI 5112 program.

b. Adjust the voltage range (e.g. 50 mV) and the time range (e.g. 500 ms)
for each channel (two channels, one for the incident bar and the other
for the transmitter bar).

c. Set up 100 mV for system triggering.

B. Power and Circuits

16

a.

b.

Turn on the power supplies for the Wheatstone bridge circuits and the
ampliﬁers.

Check all connections from the gages up to the computer.

C. Split Bars

a.

Adjust the gap between the striker bar and the incident bar. Do not use
a gap more than one inch on the ﬁrst try.

Adjust the gap between the transmitter bar and the bar stopper.

Install the specimen with the lubricant.

Close the protection cover case.

D. Gas Pressure

9.
h.

Close all valves.

Close the ﬁring switch.

Open the tank valve.

Open the 100 psi valve (if used).

Open the gun valve.

Open the Nitrogen valve to a pressure above the ﬁring pressure.
Bleed to set the pressure.

Close the gun valve.

E. Safety Caution

a.

b.

Clear the area surrounding the bar launching mechanism.

Notify personnel in the testing area of test.

F. Execution

17

a. Record test date and time, specimen material and dimensions,
operating gas pressure and the gap between the striker bar and the
incident bar.

b. Fire the gas gun.

c. Check the ﬁgures in the screen and save the data.

5.2 Shutting Down Procedures
A. Close the tank valve.
B. Close the 100 psi valve.
C. Open the gun valve, bleed valve, nitrogen valve and 100 psi valve.
D. Close all valves.
E. Turn off the switches of the power supplies.
F. Rest striker bar on a nylon pad.

G. Exit from the computer program and turn off the computer.

18

CHAPTER 3

CALIBRATIONS OF COMPONENTS AND SHPB

Once the Split Hopkinson Pressure Bar (SHPB) is constructed, calibrations
should be performed before the bar can be used for any material
characterization. The calibrations of the SHPB must be conducted on individual

components as well as the whole SHPB.

1. Calibration of Differential Ampliﬁers

During the operation of the Split Hopkinson Pressure Bar, elastic strain waves
are expected to travel through the strain gages mounted on the incident bar and
the transmitter bar. The shape of the elastic waves is usually close to square due
to the blunt impact between the striker bar and the incident bar. The frequency of
the waves should be around 3,500 Hz because the bars are made of 347
stainless steel. Hence, a square wave with a frequency of 3,500 Hz was created
by a function generator and used in the calibration of differential ampliﬁers.
Figure 6 shows the input and output waves on the oscilloscope. Although the
square corners of the waves are slightly smoothed out, the overall quality of the
output waves seems to remain very close to the input waves.

Figure 7 shows the gain (i.e. the magniﬁcation factor) as a function of
frequency. Apparently, an almost constant gain exists for frequencies up to
10,000 Hz. For a frequency of 3,500 Hz, a linear relationship between the input

and the output voltages is identiﬁed and given in Figure 8. The linear relation has

19

 

 

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Figure 6 Output wave (below) of an initially sqaure wave (top) with
a frequency of 3500 Hz from a differential ampliﬁer.

20

25 5” 7777 55 5

 

 

 

 

 

 

2K)4I~ﬂU'“‘f““8""““r"9-vr—V; e ; A
\

.E 15 i
(O
(D

10 l

0 Measured
5 ‘ —Poly. (Measured)
O r l l l
0 1 0000 20000 30000 40000 50000

Frequency (Hz)

Figure 7 Frequency response function of ampliﬁer

 

700

Soo8

 

Output (mV)

§§

 

 

 

T r

0 5 10 15 20 25 30 35
Input (mV)

 

Figure 8 Gain of ampliﬁer based on square waves at 3500 Hz

21

a slope, i.e. the gain, of 20.38. This value is very close to what was mentioned in
section 3.2 of Chapter 2. Results from both Figures 7 and 8 seem to validate the

accuracy of the differential ampliﬁers in the measurement of strain waves.

2. Calibration of Wheatstone Bridge Circuits (with Differential Ampliﬁer)

Shunt calibration is the common technique for calibrating Wheatstone bridge
circuits. It was also used in this thesis research. As shown in Figure 9, each
Wheatstone bridge circuit consists of two active gages and two dummy gages
with nominal electrical resistance of 3500. An adjustable resistor as shown in
Figure 9 is also used for initial circuit balancing.

The resistance of the calibration resistor R6 should be at least:

R = g -R (3.1)

 

where R9 is the gage resistance, i.e. 3500, F9 is the gage factor and 65 is the

strain limit of the gage. If the gage deformation limit is 3% and the gage factor is
2.105, the resistance of the calibration resistor should be at least 5.192 kn
according to Equation (3.1). Resistors of 4.67 k0, 6.78 kQ, 9.95 kQ, 21.8 k0,
46.6 k!) and 996 k!) were also selected and used to evaluate the linearity of the
output from the Wheatstone bridge circuit.

A DC power supply with a constant voltage of 4.88V was used in each
Wheatstone bridge circuit. In calibrating the circuits, the resistors were shunt one
by one and the corresponding voltage from the oscilloscope was recorded. The

simulated strains can then be calculated from:

22

 

   
     
 
     
   

996 k ,
Dummy Gage

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

46.6 k ,
21.8 k 350 Q
9.95 k
6.78 k ctive Gage
4.67 350 Q DC 4.88V
Active Gage
350 Q
Adjustabl V
Resistance 0

0~25 Q
I” """""""" l """ '1
I I
l I
l I
I I
i i :1
: : _
I m
I I
i l o
l I
I I
L---------__..------.' ,-

Differential
Ampliﬁer Computer Based
Oscilloscope

Figure 9 Wheatstone bridge circuit and shunt calibration resistors.

23

0.04 -5——<7~~—5—~ 77,, ,,7,

0.03 -‘

.O
o
N

Experimental Strain
O
'2

(shunting Rg on Rc)

 

 

 

 

10.0%
8.0%
6.0%
4.0%
2.0%
0.0%

-2.0%

-4.0°/o

-6.0°/o

-8.0%

-10.0°/o

Error Percentage %

l

0.01 0.02 0.03 0.04

 

 

 

Theoretical Strain
(a)
o
F i o i
o 0.01 0 0.02 0.03 0.04

 

 

Theoretical Strain

(b)

Figure 10 (a), (b) Comparisons between experimental measurements
and theoretical calculations.

24

R

g
as = (3.2)
Fg (Rg + RC)

 

The direct measurements and the simulated strains were compared and are
shown in Figure 10. They seem to agree with each other very well, validating the

accuracy of the Wheatstone bridge circuits.

3. Calibration of the Whole SHPB
Based on the theory of wave propagation, the wave speed in a material Co,

the dynamic Young’s modulus of the material E and the density of the material ,0

have the following relation:
C, = E (3.3)
,0

The wave speed can also be identiﬁed from the wave pattern based on the
following formula:

21
Co = — 3.4
T < )

where 21 is the total length of wave propagation and T is the traveling time of the
wave within the distance 21. Both Equations 3.3 and 3.4 are useful for the
calibration of the whole SHPB. In this thesis research, the following three tests

were designed and performed for the calibration of the whole SHPB.

3.1 Calibration with Load Cell
In this calibration study, two strain gages were mounted on the striker bar

close to the impacting end. The installation of the strain gages and the

25

subsequent circuits design were identical to those used for the incident bar and
the transmitter bar. In addition, a calibrated load cell, usually used for impact
testing, was placed in front of the striker bar as shown in Figure 11. When the
striker bar impacts the load cell, the history of the contact force can be detected
by the load cell and recorded in a computer. Similarly, the history of the strain
waves can be detected by the strain gages on the striker bar and recorded in a
computer.

Based on the recorded strain wave, the wave speed in the striker bar can be
calculated with the use of Equation 3.4. Subsequently, the dynamic Young’s
modulus E of the striker bar can be determined by using Equation 3.3, the
calculated wave speed and the density of the striker bar ,0, i.e. 489.6 lb/ft3
(7,859.1 kg/ma). For example, if the wave speed is 16,188 ft/s (4,934.2 m/s), the
dynamic Young’s modulus of the striker bar E will be 27.3 Msi (191.4 GPa). This
value is not very much different from the static Young’s modulus — 27.6 Msi (193
GPa).

The stresses in the striker bar can be identiﬁed from two independent
methods: the load cell and the strain gages on the striker bar. Based on the load
cell, the stresses can be obtained by dividing the contact forces with the cross-
sectional area of the striker bar. Based on the strain gages, the stresses can be
obtained from the multiplication of Young’s modulus with the strains. The
procedure to determine the Young’s modulus has been mentioned earlier. The

procedure to determine the strains is given below.

26

Striker bar with strain gages

 

 

 

 

 

 

Load cell

 

 

 

 

 

Wheatstone

Bndge

 

 

 

 

 

Ampliﬁer

 

 

 

 

 

 

 

 

 

 

/

 

 

 

,:
O
L"

 

 

 

/

l

 

1

Computer Based
Oscilloscope

 

 

 

 

 

 

 

 

 

7

EB

 

 

 

 

 

Dynatup 830 dynamic force
measurement system

 

Trigger the two systems
at the same time and

set proper delays.

 

 

 

Figure 11 Schematic diagram of striker bar impacting on load cell.

27

The change of voltage in a Wheatstone bridge circuit due to the changes in

strain gages can be expressed as

EC*F

AE 4 g (‘51 + £2 — £3 + £4) ' (3.5)

where AE is the change of voltage, Ec is the circuit voltage (4.88 V) and F9 is the
gage factor (2.105). Since AE is a reading from the ampliﬁer, it should be divided
by the gain (20.38) before being substituted into the above equation. Equation
3.5 can be further simpliﬁed as

an:
EC Fg

if there are no strains in the dummy gages.
Figure 12 shows the comparison of stresses from the load cell and the strain
gages on the striker bar. The results from the strain gages seem to be constantly

lower than those from the load cell. The offset is around 286 psi (2 MPa). A

calibration for the load cell may be required.

3.2 Calibration with Instrumented Striker Bar

In this calibration study, the instrumented striker bar used in the previous
section was also used to impact the incident bar, the transmitter bar and the
combination of the two bars, shown in Figures 13 (a-c), respectively. As
mentioned earlier, the dynamic Young’s modulus of the striker bar can be
determined from the wave propagation equation, Equation 3.4. Similarly, the

Young’s moduli of the incident bar and the transmitter bar can be characterized

28

O)
01
l
l
l
l
l
l
l
l
|

 

 

O)
O
l

01
01
l

01
0

Equal Line

.h
0

Stress from strain gages in striker bar (MPa)
no A
O1 01

OD
O
i

 

N
01
l

    

 

 

 

l l l

25 30 35 40 45 50 55 60
Stress from load cell (MPa)

Figure 12 Comparison between the signals from load cell and
those from strain gages in striker bar.

29

65

' ' h tl '
striker bar wt 5 ram 93995 Incident bar (a)

 

 

 

 

 

7:};'=-=i?

 

 

 

 

 

AL AL

striker bar with strain gages Transmitter bar

('0)

 

 

 

 

 

 

 

 

 

 

 

AL KL

striker bar with strain gages
Incident bar Transmitter bar (C)

l l

 

 

 

 

 

 

 

 

 

l J I l l I 1 l [I
Figure 13 (a-c) Schematic diagram of the striker bar impacting the

incident bar, the transmitter bar and the combination of
the two bars.

30

with the same method. The density, wave speed and Young's modulus of each
bar are summarized in Table 1. They are all very close.

in order to further compare the discrepancy of measurements between the
bars, measurements of strains due to the impact between the striker bar and the
incident bar, and that between the striker bar and the transmitter bar, were
performed. The experimental results are given in Figure 14 for comparison. The
strains in the incident bar and those in the transmitter bar are normalized with the
strain in the striker bar. The discrepancies of strain measurements between the
incident bar and the striker bar (45° line) and that between the transmitter bar
and the striker bar (45° line) are around 8% and 6%, respectively. Misalignments

of bars and strain gages are believed to be responsible for the discrepancies.

Table 1: Density, wave speed and Young’s modulus of split bars.

 

 

 

 

Density (kg/m3) Wave Speed (m/s) Young’s Modulus
(GPa)
Striker Bar 7855.03 4935 191 .245
Incident Bar 7858.03 4935 191 .376
Transmitter Bar 7855.85 4946 192.1 80

 

 

 

 

 

 

Besides the accuracy, the range of application is also an important concern in
the calibration of SHPB because SHPB will be used at various levels of high
strain rate. In this calibration study, the wave speeds of the individual bars
subjected to various levels of impact pressure were investigated. Figure 15

shows the experimental results. They seem to indicate that the wave speeds,

31

03
O
1

01
01

01
O
1

 

/

y = 1.0833x - 1.7755 12’

l

/
AA /
’l

/

y = 1.0625x - 2.0904

(mV)
8:

A
O

(JD
(11

00
C

Incident Bar Signal & Transmitter Bar Signal

 

// Equal Line

A I vs S
El T vs 8
Linear (I vs S)
---— Linear (T vs S)

 

 

N

U1
N
01

l l l l l

35 40 45 50 55
Striker Bar Signal (mV)

Figure 14 The discrepancies of strain measurements between
the incident bar and the striker bar and that between
the transmitter bar and the striker bar.

32

 

5100

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 e——‘—v f ; e t t H
9’ s
2 ‘E’ 4700
5 (a)
4500 , I . 4
5 15 25 35 45 55
Pressure (psi)
5100 7 -- «mun—77 , 7
-o o
0 0—T f : z
a 1.; 4900 -
a) L
g g 4700
5 (b)
4500 , T 1 1
1O 20 30 4O 50
Pressure (psi)
5300 5-
8 5100 — .
g. A Q
m t
‘3 E 4900 ~ .L—T T ,
> v
9 4700 ~ (c)
4500 , . . T

Pressure (psi)

Figure 15 Wave speed as a function of gas pressure.

(a) Striker bar impacting the combination of incident and
transmitter bars.

(b) Striker bar impacting the incident bar.
(0) Striker bar impacting the transmitter bars.

33

and hence the Young’s moduli, of the bars are not sensitive to the impacting
pressure up to 360 psi (2.52 MPa). The attenuations of the strain waves in the
incident bar and the transmitter bar were also investigated. Figure 16 shows the
results displayed on the oscilloscope. The time durations for the strain waves
with 50% of amplitude reduction are given in Figure 16 for comparison. A
noticeable difference between the incident bar and the transmitter bar seems to

exist.

3.3 Calibration by Testing 6061-T6 Aluminum

The most common approach to calibrate an SHPB is to characterize a well-
known material. Results from the characterization can be used to evaluate the
SHPB when compared with the existing knowledge. In this study, 6061-T6
aluminum was used for the calibration of the SHPB. Cylindrical specimens were
prepared. Although suggestions on the specimen dimensions and dimensional
ratios are available in the literature, there is no conclusive suggestion. In fact, the
dimensions and dimensional ratios used by different researchers vary a great
deal. Hence, specimens with various dimensions and dimensional ratios were
investigated. Figure 17 shows the typical incidence, reﬂection and transmission
waves recorded by the bars.

In order to understand the wave propagation measured from the SHPB, it is
useful to examine a schematic diagram of time-positon relation of the strain
waves in the incident bar, specimen and transmitter bar, as shown in Figure 18.

The waves in the striker bar and the incident bar are generated right after the

34

 

Striking bar impacting on transmit
bar at gas pressure of 275 kPa

 

 

 

 

Striking bar impacting on transmit ,
bar at gas pressure of 103 kPa ' I ;

I

l

 

 

 

 

 

 

 

 

 

 

1213469 vﬁlg—ZIWOEJI—H -3" 5.: 518-3- .1713” >hlgrfE—H "‘32! .55 5 3513—:-
( c ) Striker bar - Incident bar

A 150 5 5 *5 -
‘é’
V , 0
c 100 0 3 3
.2
E 50
3
o

0 . , T 7

0 10 20 30 40 50 60
Pressure (P.S.l)
( d ) Striker bar - Transmitter bar

... 150
"’ 8
E
V 100 5
§ ‘ I o
E 50 —~
3

O l l l l

0 10 20 30 40 50 60

Pressure (P.S.l)

Figure 16 (a), (b) Strain waves showed on oscilloscope during 500 ms
(0), (d) Time durations for the strain waves with 50% of amplitude

reducﬂon

35

 
 

JVqualBenchrScope . I NI 5112 lor PCI ﬂ

.1,qu
Voﬂs/Div Ref 1: 50 mV Rel 2' 50 mV

Aluminum 6061-T6 D12.6L25.38 Gas pressure: 380 psi

T l

Reﬂected
Strain Signal

 

 

 

 

 

 

 
 
  
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I

l Incident

i—m— Strain Signal 7,,7 7
I
l

/

5 .
| I
l I
An 4A ‘_ A :J In“ A u; A AA u an; IL A u m ‘ALlA-‘ ‘1
v1 WV 1 WT T v v v "v 1 v vvvv v v w - >—-— -———-- -—
I

 

 

T77 _ ,

5 L +4 77 Transmitted —»—-——§
. . 1 If , -'
L Strain Signal j ‘

j l _ l w l l, ,, Fri“

 

 

 

 

 

 

 

 

 

53m 53.4"» "

.1 £15,:- '
9mg] .swefirfj . . 5:3,: )‘il "mfg .. swéieil
asmlll [A a 3;: b F ”I-Virhaaﬂoneh-Scopo... - - 5

 

 

’ " ~ 5 ' . . ‘6'».
h at. M - .I'.-. I .o . -f_- \:.q ... j. , -r"
- a

Figure 17 Typical incidence, reﬂection and transmission waves from
SHPB for Aluminum 6061-T6.

36

For an Al 6061-T6 specimen (¢12x15mm), the
wave will go through and reﬂect in the

 

 

 

 

 

 

 

 

 

 

 

Time (uS) specimen roughly 15 times.
4
600 —
500 — :
400 -— : I
300 — 5 8r ,, -‘ : St
200 — E ', , E
: at M E
100 - : 5
o ' '
Striker Gages 419 mm 419 mm hages
I\ /II\ /I
|/203 "“314 838 mm \II/ 838 mm
l‘ ’l‘ ’ll‘

Figure 18 Schematic diagram of time-positon relation of the strain
waves in the incident bar, specimen and transmitter bar

37

impact between the striker bar and the incident bar. The wave in the striker bar

propagates to the free end of the bar while the wave in the incident bar, the so-
called incident wave, i.e. 5,, propagates to the interface between the incident bar

and the specimen. The length of the incident wave is proportional to the duration
of the imapct-contact. It ends when the striker bar separates from the incident
bar. Based on the theory of wave propagation, the separation is due to the arrival
of the tensile wave on the contact surface between the two bars. This happens
when the impact-induce compressive wave in the striker bar returns from the free
end of the striker bar. Accordingly, the incident wave is an input wave for material
characterizations at high strain rates. It is noted that the incident wave is only
based on the impact-contact relation between the striker bar and the incident bar.

Once the incident wave arrives at the interface between the incident bar and
the specimen, one part of the strain wave continues to propagate into the
specimen while the other part is reﬂected from the interface. The reﬂection part

will again be dectected by the strain gages mounted on the incident bar and is

called reﬂection wave, i.e.8R. The division of the incident wave into a

propagation part and a reﬂection part is essentially based on the mechanical
impedances, i.e. the Young’s moduli, of the specimen and the bar. If the
specimen has a higher mechanical impedance than the bar, the strain wave will
experience larger propagation than reﬂection. On the contrary, if the specimen
has a lower mechanical impedance than the bar, the strain wave will experience
higher reﬂection than propagation. The difference between the propagation part

and the reﬂection part should not be too great if an accurate measurement is

38

desired. For example, a signiﬁcant error may arise when using a high-modulus
SHPB to characterize a low-modulus polymer.

As the wave in the specimen continues to propagate, it will soon hit another
specimen-bar interface. Another wave division into propagation and reﬂection will
take place on the interface between the specimen and the transmitter bar. The

part that propagates into the transmitter bar will be recorded by the strain gages

mounted on it and is usually called transmission wave, i.e.é‘T. The part that

reﬂected from the interface will propagate through the specimen before it is again
divided into propagation and reﬂection parts at the interface between the
specimen and the incident bar.

As mentioned earlier, the duration of the incident wave is proportional to the
double-length of the striker bar. Since the thickness of the specimen is much
shorter than the double-length of the striker bar, many wave divisions may take

place on both interfaces within the duration of the incident wave, also shown in

Figure 18. As a consequence, both the reﬂection wave ER and the transmission

wave 6‘, will be continuously modiﬁed with the addition of the wave components

propagating through the specimen. Apparently, the higher the number of times
the wave bounces between the two ends of the specimen, the greater the
inﬂuence of the specimen properties on the waves. In other words, the many
wave bouncings between the two ends of the specimen will impose a more
uniform wave propagation to the specimen, and hence a more uniform

deformation, which is an important assumption of the SHPB technique.

39

The incidence, reﬂection and transmission strains shown in Figure 17, i.e. 8,,

ER and 6}, respectively, are required for establishing the stress-strain curve of

the specimen under investigation. Because these waves do not take place at the
same time, it is important to carefully match the initial points of the waves to
establish accurate results. In this calibration study and subsequent studies,
details of the strain histories were examined. Artiﬁcial judgements were made to

determine the starting point of each strain wave.

A. Input and Output Force History

An important step to validate the strain waves is to compare the input force
history and output force history at the two ends of the specimen [3]. The former is
the multiplication of the difference of strain history between the incidence wave
and the reﬂection wave, recorded by the incident bar, with the Young’s modulus
while the latter is the multiplicaiton of the strain history of the transmission wave,
recorded by the transmitter bar, with the Young’s modulus. Figure 19 shows the
results from four specimens with different diameters and lengths and subjected to
a gas pressure of 400 psi. The dimensions and dimensional ratios of the
specimens are summarized in Table 2. Apparently, the discrepancy increases as
the ratio between the specimen diameter and the bar diameter increases. The

result seems to suggest the diameter of the specimen should not be too small.

40

 

015.88 L12.86-400 psi

       
 

 

  
  
       

    

 

 

60 1
50 4
g 4o~
8 30 —
E 20 4 —Output force
10 I —Input force
0 r t .
0 0.05 0.1 0.15
Time (ms)
D10.68 L10.4-400 psi
40 1
E30 5
5
@320 z
2
no. 10 4 —Output force
—lnput force
0 l l I
0 0.05 0.1 0.15
Time (ms)

Force (kN)

 

 

 

 

012.3 L13.3-400 psi
50 -
A 40
2
i‘. 30 ~
o
2 20 .
LE —Output force
10 L —lnput force
0 l I 1
0 0.05 0.1 0.15
Time (ms)
D8.08 L8.04-400 psi
30 1
20 1
10 ~ ‘
—— Output force
—lnput force
0 I 1 ﬂ
0 0.05 0.1 0.15
Time (ms)

Figure 19 Input and output force histories of four aluminum
specimens with the same impact gas pressure.

41

Table 2: Dimensions and dimensional ratios of aluminum specimens.

 

 

 

 

 

 

 

 

 

 

 

D: Diameter L: Length L/D D/Dbar
(mm) (mm) Ratio Ratio
Specimen A 15.80 12.86 0.81 0.84
Specimen B 12.80 13.30 1.04 0.67
Specimen C 10.68 10.40 0.97 0.56
Specimen D 8.08 8.04 1.00 0.43
Dbar= 19 mm

D: specimen diameter

B. Stress-strain Curve

Since SHPB is a one-dimensional technique for material characterizations, it
is ideal that uniform stress, strain and strain rate take place in the specimen
during the testing. In order to establish the stress-strain curve for the specimen
material, it is necessary to investigate the displacements and forces at both ends
of the specimen. If the forces, and hence the stresses, on the two ends are
approximately equal to each other, the stress in the specimen could be
considered to be uniform in the specimen and the stress and strain histories can

be determined by using the following equations based on the transmission and

reﬂection waves, i.e. a, and 6R , respectively,

_ EAB
As

0'

_ CO I 1
a _ —2 520—8” (1 )dt

 

42

(3.7)

(3.8)

 

where E is the Young’s modulus of the transmitter bar, A3 and A3 are the cross-
sectional areas of the bar and the specimen, respectively, Co is the wave speed
in the bar, and L0 is initial length of the specimen.

In presenting a stress-strain curve, it is useful that the stress-strain relation is
based on a constant strain rate. The strain rate can be obtained from taking the

time derivative of the above strain equation, i.e.

3——2§£g (3 9)
L R '

0

Since the reﬂection strain 6,, is generally a function of time, it is imperative that

constant strain rate is achieved during the test. OthenNise, special techniques,
such as shaping technique for incident wave, are required to obtain stress-strain
curves at various strain rates.

Figure 20 shows the stress-strain curves from six tests. They were obtained
from Specimens with different dimensions and strain rates. The notation used in
the diagram represents the diameter D and the length L of the specimens. For
example, D12.6L35.38 denotes a cylindrical specimen with a diameter of 12.6
mm and a length of 35.38 mm. The value given behind the hypen denotes the
strain rate in terms of s". The result from Chen [5] is also given for comparison.
Table 3 shows the yielding points obtained from all tests. The discrepancies
between Chen’s study and this calibration study are likely caused by differences
in both dimensions and strain rate. Chen’s result was obtained from a relatively
uniform strain rate after using a shaping technique to impose the relatively

uniform reﬂection wave while this calibration study was based on relatively

43

Stress (MPa)

 

 

 

450 5
400 — //

— D12.6L25.385-290 1/s
—— D12.44L19.12-330 1/s

I ------ D15.88L12.86-420 1/s

- — ~ ~— D12.8L13.3-540 1/s

— - - — D10.68L10.4--700 1/s

— — —- - D8.08L8.04-1050 1/s
__ Chen[5]-830 1/s(const)

 

l

l l l l l

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Strain

Figure 20 The stress-strain curves of aluminum 6061-T6
under different strain rates.

44

constant stresses, i.e. relatively uniform incident waves. The maximum strain rate

at each calibration test was deﬁned as the strain rate.

Table 3: Comparison of Yielding Points

 

 

 

 

 

 

 

 

Specimen Yielding Point
ksi MPa
D12.6L25.38 32.1 225
D12.44L19.12 32.9 230
D15.88L12.86 30.0 210
012.8L13.3 30.7 215
D10.68L10.4 30.7 215
D808L8.04 30.0 210
Chen’s[5] 47.9 335

 

 

 

 

 

45

CHAPTER 4

CHARACTERIZATION OF THICK LAMINATED COMPOSITES

As more laminated composites are used for structures subjected to high
strain rates, such as ballistic impact and crashworthiness, the behavior of
laminated composites at high strain rates is becoming a primary concern in
composite structural designs. Since SHPB has been used as the standard
method for characterizing homogeneous materials, such as conventional metals,
at high strain rates, the feasibility of using SHPB for characterizing
inhomogeneous laminated composites has become a primary interest in
composite research. Besides the inhomogeneity, the use of the SHPB-based
characterization technique for laminated composites is becoming more
sophisticated, if not more controversial, as the laminated composites for
structural designs are becoming thicker. This thesis research explores the

application of SHPB in thick laminated composites subjected to high strain rates.

1. Speceimen Preparation

In this thesis research, laminated composites made of glass ﬁbers and an
epoxy matrix were of interest. All composite plates were cross-ply laminates and
had the following stacking sequence: [0/90/0...]17, [0/90/0...]33 and [0/90/0...]67.
They had the following thickness: 6.25mm, 12.5mm and 25 mm, respectively.

Since specimen dimensions and dimensional ratios had strong effects on the

46

behavior of composite plates, several specimen dimensions and dimensional

ratios, as shown in Table 4, were used in the study.

Table 4: Dimensions and dimensional ratios of Glass/Epoxy specimens.

 

 

 

 

 

D: Diameter L: Length L/D Ratio D/Dba, Ratio
(mm) (mm)
Specimen A 15.80 25.50 1.61 0.83
Specimen B 15.80 12.95 0.82 0.83
Specimen C 15.80 11.50 0.73 0.83
Specimen D 12.40 12.80 1.03 0.65

 

 

 

 

 

 

 

The preparation of the specimens for SHPB characterizations required extra
care. In order to prepare cylindrical specimens, a bi-metal hole saw, as depicted
in Figure 21, with different diameters were used. Because of the inhomogeneity
of the laminated composites, surface chipping and interfacial delamination can
easily take place during the machining of specimens. Hence, a proper drilling
speed and a well clamping for specimens were critically important to prepare
damage-free specimens. A drilling speed of 1000 rpm with water cooling was
found to be suitable for the glass/epoxy composite plates. Since bi-metal hole
saws with inner diameters of 15.8 mm, 12.8 mm, 10.68 mm and 8.08 mm were
available, only limited specimen dimensions and dimensional ratios were
possible. Among the combinations of specimen diameter D (based on the size of
bi-metal hole saws) and specimen length L (based on the thickness of laminated

composites), the selected dimensions and dimensional ratios are summarized in

47

 

Figure 21 Bi-metal hole saw

48

Table 4. The selection was based on the guidelines that the D/Dbar ratio should

be higher than 0.65 and the L/D ratio should be higher than 0.7.

2. Experimental Results

Figure 22 shows the typical incidence, reﬂection and transmission waves for a
laminated composite specimen. They resemble those for an aluminum specimen
up to some extent. The stress-strain curves of all tests are summarized in

Figures 23-28.

2.1 Strain-Rate Effect

Figure 23 shows the stress-strain curves for specimens with a diameter
around 15.8 mm and a length around 25.5 mm, resulting in a U0 ratio of
approximately 1.6. The strain rates vary between 190 s'1 and310 5". Generally
speaking, the stress-strain curves have similar shapes. At constant strain level,

the stress rises as the strain rate increases.

2.2 Laminated and Assembled Composites

Figure 24 shows the stress-strain curves for specimens subjected to the
same level of strain rate. Although all specimens have about the same
dimensions and dimensional ratios, they can be divided into two groups; one
group of specimens is made of laminated composites while the other group is
made of assembled composites. The laminated composites were virgin

composites with thickness around 25.5 mm while the assembled composites

49

HVIIIUdIBtnCh Scope I NI 511210114]

 

 

 

 

 

 

 

 

 

 

 

  
  
  
 
   

 

 

 

 

 

 

vapor. Flofl somv FleIZ somv ' 5 A “-5 7. '5 M91" M5 55 55 5
Position 40mV ~501nV 5 Dill L IRdZ a
Stopped 331u5ec/d1v :- 0“ I’M
') L L Reﬂected f; “5:: L j .
1 ) , Strain Signal )‘7—- m
) . . Al L953
r ) -- --;.......——- ......
+ 1) l)
l [lBHf34mammﬁB

 

     

 

IQFMlél ﬂdhmﬁa)

. _ . . . , L 7-) _,__._ _ ,» _. 7 .. 1
Incident L I S" [,1 El) Sheilf
Strain Signal 4 3 ‘ [Lt/Im\ L A ﬁZLiL . “Fm—Ll )

 

' 5 L \ l 5' L '

L L L 7 L, / L \‘N-t I L

) -74.. , . 7 n 7‘ J i\ )

v_. )7 .7 1,7,. 777i, 7, l
L L L

 

 

 

, . Transmitted
L 5 Strain Signal

 

 

 

 

 

 

 

53,6111: 5 L L ' 540m: 5 .5 L

. M .lﬂ—I. I 1
811th -7 SW22] Scull] ._ ._ Hed- ~31 I Mg L ,, j
351ml} :1] 6 :31 r 7‘ goon-mi mmwﬁﬂgvmmsm... 1% 111mm“

 

Figure 22 Typical incidence, reﬂection and transmission waves from
SHPB for glass-epoxy composite

50

250 -)

    

 

 

 

 

200 ~-
13 150 l
CL
E
(I)
(D
Q i
..'=
(I) 100
/ ‘ — -—D15.7L25.18—2601/s—-1
v — - — D15.7L25.46-2401/s—1
5 ’ —— D15.8L25.06—290 1/s-2
50 ii/ , --~~——D15.8L25.56—250 1/s—2
LIL ------ D15.8L25.54—310 1/s—3
i — —— D15.8L25.6—1901/s-3
O U " I ‘ l i W
0 0.005 0.01 0.015 0.02
Strain

Figure 23 Stress-strain curves for glass-epoxy composites
with similar dimensions and U0 ratios (1.6 ).

51

 

 

250 —
200 i _
T6 150 5
O.
E
ll)
(0
d)
z
m 100 5
) —— — Dl5.7L25.18-2601/s—1
) . — -— D15.7L25.46—240 1/s-1
_ - —— - — D15.8L25.56—2501/s-2
50 ' ------ D15.8L12.42)Q—2501/s—1

 

)2"
——~ - D15.8L12.55)Q—240 1/s—1

T

O —;— 1 i
0.005 0.01 0.015 0.02

Strain

Figure 24 Stress-strain curves for two groups of glass-epoxy composites
with similar strain rates and total dimensions.

Group one—from thick glass-epoxy composite plates,
Group two—from thin glass-epoxy composite plates and assembled

52

were made from placing together two thin laminated composites, each had a
thickness around 12.5 mm. The lubricant applied to the interfaces between the
ends of specimens and bars was also applied to the interfaces between the thin.
laminated composites for the ensurance of continuity of wave propagation. From
Figure 24, it can be seen that the laminated composites have slightly higher

stresses than those assembled at the same strain level.

2.3 Effect due to LID Ratio

Figure 25 shows the stress-strain curves of specimens subjected to about the
same strain rates. However, the LID ratio of the specimens are not the same; two
specimens have the LID ratio greater than 1.0 while the third one has the LID
ratio around 0.8. The difference of stress-strain curve caused by the difference of
LID ratio is very signiﬁcant. With the same strain level, the specimens with larger
LID ratios have higher stress than the specimen with a lower LID ratio. A similar

result can also be found from Figure 26 for another study.

2.4 Effect due to Specimen lnhomogeneity

Figure 27 shows a study similar to the previous section. Once again, there
are two LID ratios; one is 1.62 and the other is 0.82. The specimen with the
higher LID ratio has higher stress than those with the lower LID raito when the
specimens are compared at the same strain level. It should be pointed out that
the specimens with the lower LID ratio were prepared from the material with the

higher LID ratio by removing half the layers from one side, namely the edge

53

specimens. This exercise is based on the fundamental understanding that the
material properties in a thick laminated composite are not uniform through the
laminate thickness due to the nonuniform thermal curing cycle throught the
laminate thickness. It is believed that the-composite layers close to the laminate
surface have higher material properties than those close to the midplane due to
the imposition of a more accurate curing cycle on the layers close to the lamiante
surface. Figure 27 seems to indicate that the effect due to LID ratio outplays the
effect due to the difference in material properties.

In order to further verify the effect due to the property difference, more tests
were performed and the results are given in Figure 28. All specimens have about
the same level of LID ratio. Three types of specimen were used in the study.
Those designated with “edge” and “middle” were prepared from laminated
composites with 67 layers, i.e. the [0/90/0...]57 compsoite plates. The “edge”
specimens were prepared from removing about 33 layers from one side of the
laminated composites while the “middle” specimens were prepared from
removing about 27 layers, from each side of the laminated composites. The
specimens “without” any designation were virgin laminated composites, which
had 34 layers, i.e. [0/90/0...]34 compsoite plates. Experimental results clearly
shows the rising trend of the stress-strain curves from “middle” to “without”, then

to “edge” under constant strains

2.5 Young’s Modulus and Yielding Point

54

300

 

 

 

250 ‘1,
200 “
70‘
n.
E I
m 150 T
(D
g /,I
m : /
100 //

/ ,
) // " ,, —- ——D12.4L12.8—490 1/s—3
50 )x // ------ 012.3L12.76-4401Is-3
/ ./
/

ll , ———- 015.8L12.7-470 1/s—3

O

 

l l l l

0 0.005 0.01 0.015 0.02 0.025
Strain

Figure 25 Stress-strain curves for glass-epoxy composites
with similar strain rates but different LID ratios.

55

300 ‘i
/—‘\
/
250 /(,gx\
7 ,' \\

200 '1 ,7 I //
I/ I, III/ 1' I.
,1 /, /

150 ‘ ,
i ///, '
/’/' / [ﬂ 1' ,’ /
// ,{4' ,z ‘— — D12.4L12.8—490 1/s—3
100 - 4,7.” / ------ D12.3L12.76—4401Is-3
/ ' '
(é). —— D10.84L12.14-600 1/s—3
' - ——D10.84L11.28-470 1/s—3
-5 -. -— - — D15.8L12.6—-570 1/s-1
50 1,, f, .
; — - - — D15.8L12.4—-5401/s—2
- - - - D15.8L12.7—4701Is-3

0.03 0.035

Stress (MPa)

 

I’yL
l __ l l

0 ‘l l l
0 0.005 0.01 0.015 0.02 0.025
Strain

Figure 26 Stress-strain curves for glass-epoxy composites
with different strain rates and different LID ratios

56

250 )

Stress (MPa)

 

 

 

200
150 - , - \
l
I
,/'
100 . ,/
50 5 If»? '/ —D15.8L25.54—3101Is—3
)L / — - - -— D15.8L12.96-edge—3401/s—-3
lI ------ D15.8L12.94—edge—320 1/s—3
o . . . —,
0 0.005 0.01 0.015 0.02
Strain

Figure 27 Stress-strain curves for glass-epoxy composites
with similar strain rates but different LID ratios.

57

250 )

 

Stress (MPa)

 

 

 

200 5
150
100 5 ; 5 . ’
/’ , .
/ .545/ (.5, - - ——D15.8L11.90-edge—570 1/s-1
/ r
43/ ' .' -—-—D15.8L11.44-edge—570 1/s-1
571;? ' — ----- D15.8L11.64-edge—570 1/s-1
5O 5 g’,‘ /;'5 " D15.8L12.6—5701/s—1
’ ’2; .
If Dl5.8L12.4-540 1/s—2
I"’ ------ D15.8L11.4-middle-550 1/s-1
,‘,l
~ 5 5 5 D15.8L12.5-middle—5401/s—1
In"
0 ./ 7 l r l ‘1
O 0.005 0.01 0.015 0.02 0.025

Strain

Figure 28 Stress-strain curves for three groups of glass-epoxy composites
with similar dimensions and LID ratio (0.8)

58

In addition to the stress-strain curves, Young’s moduli and yielding points of all
cases presented earlier were also carefully examined. The results are shown in
Figures 29 and 30. From both ﬁgures, Young’s modulus and yielding point
increase as the strain rate increases. They also increase as the LID ratio

increases.

59

Yielding Point (MPa)

 

 

 

 

 

 

 

0
14 I o o
O o D D
120 I D E]
) °o o lJD=1.6
100 El LID=1
80 5 A lJD=0.8
x UD=0.8 (edge)
60 i X x UD=0.8 (middle)
x)< A 8
40 5 A 15K
20 5
0 l l l 1
0 200 400 600 800
Strain Rate (1Is)
Figure 29 Young's Modulus vs. strain rates at different UD ratios.
80 ..--_ A W~“_ E, 7 L 7#
70 5 x
0 0° x
60 ‘ <9 (:1 El A x
50 ~ D A
o A o LID=1.6
40 ‘ xx xx :1 LID=1
30 . A UD=0.8
x UD=0.8 (edge)
20 “ x UD=0.8 (middle)
10 )
0 . T T
0 200 400 600 800

Strain Rate (1Is)

Figure 30 Yielding points vs. strain rates at different L/D ratios.

60

CHAPTER 5

DISCUSSIONS

The application of SHPB to material characterization is constrained by the
material properties and dimensions of the bars and the specimen. As a general
rule, the material properties of the specimen should be softer but not too much
softer than those of the bars to warrant relatively equal measurement between
the reﬂection and transmission waves, which is necessary for establishing an
accurate stress-strain curve [1, 3]. In addtion, from the above studies for
laminated composites and the calibration studies for aluminum, it can be found
that the dimensions and dimensional ratios of a specimen have strong effects on

the stress-strain relations based on SHPB. Hence, they can not be ignored.

1. Dimensions of Bars
1.1 Bar Diameter Dba.r

SHPB is a one-dimensional wave technique for material characterization.
When testing materials are homogeneous, specimens with small diameters will
be adequate. Consequently, incident and transmitter bars with small diameters
will be sufﬁcient. The determination of the bar diameter will then be dependent on
the desired strain rate since it can be increased by reducing the specimen
diameter. In contrast, when testing materials are not homogeneous, specimens
with large diameters will be required. If the specimen is not uniform, it will not

reﬂect the whole structure when the specimen is very small. Therefore, the bars

61

for characterizing inhomogeneous materials, such as concretes, usually have a

diameter of 3” (75 mm) or greater.

1.2 Bar Length Lba,

The ﬁrst rule to determine the length of the incident and transmitter bars
requires that the bars should be long enough to warrant one-dimensional wave
propagations. In other words, the transverse inertia should be negligible when
compared with the longitudinal counterpart. An aspect ratio, LbarlDbar, around 10
is usually agreed by researchers [2] as a minimum requirement for a one-
dimensional bar.

As the striker bar impacts the incident bar, the wave generated in the latter is
usually highly complex due to the nonuniform contact between the striker bar and
the incident bar, and the non-longitudinal waves, such as spherical dilatational
wave, generated from the impact. Although Saint Venant’s principle states that
nonuniform contact force can be quickly dampened out in a distance
approximately equal to bar diameter into the bars if the bar are made of isotropic
materials, the non-longitudinal waves may take 10 times the bar diameter to
vanish. Moreover, if electrical-resistance strain gages are to be installed in the
bars for wave measurements, the reﬂection waves from each bar end should be
considered in determining the bar length. A bar length more than 20 times the
bar diameter (10 times from each bar end) may be required to avoid the overlap

of propagation waves with reﬂection waves.

62

'07: z

..p.

 

 

Although longer bars seem to be excellent for one-dimensional wave
measurements and can help to reduce the end effect and transverse effect, they
also pose serious challenges on the straightness of the bars. As a bar becomes
longer, it sags due to its own weight. The curvature of the bar, accompanied by a
low critical buckling load, will have a negative impact on one-dimensional wave

propagation.

2. Dimensions of Specimens
2.1 Specimen Diameter D

The diameter of specimens, D, should always be smaller than the diameter of
bars, Dbar, to warrant complete wave propagations into and from the specimens.
This is especially true for specimen materials with positive Poisson’s ratios. They
expand transversely when subjected to longitudinal compression. A DID)Jar ratio
around 0.75 is commonly accepted by SHPB researchers [1, 2]. The calibration
studies for aluminum specimens in this research have also conﬁrmed that the
discrepancy between the input and output force histories increases as the D/Dba,
ratio decreases. In other words, the wave patterns seem to be distorted more

when the DIDMr ratio decreases.

2.2 Specimen Length L
Many researchers have investigated the effects of the U0 ratio. No deﬁnitely
idea ratio has been concluded from their studies. The selection of an LID ratio,

however, may be based on the following arguments.

63

(1) One of the fundamental assumptions of the SHPB technique is the
uniform deformation of the specimen during characterization; however, it is more
difﬁcult for a longer specimen to meet the uniformity requirement than a shorter
specimen.

(2) In fact, a longer specimen also suffers lower stability when subjected to
compressive loading.

(3) The stability requirement, however, cannot be satisﬁed by simply using a
very short specimen because of the following considerations: (a) one-
dimensional assumption (transverse inertia), (b) Saint Venant’s end effect
(impact-induced contact), and (c) the frictional effect due to Poisson’s effect.

Table 5 shows the suggested LID ratios from various researchers. Follansbee
[2] suggests that the optimum UD ratio to minimize errors due to inertia is 0.5.
ASTM standard E9 speciﬁes that L/D ratio should be from 1.5 to 2.0. Dowling [1]
has suggested choosing an LID ratio of 3 for ductile materials and from 1.5 to 2

for brittle materials. Apparently, there is no conclusive suggestion.

3. Constant Strain Rate

The strain rates based on constant incident strain wave for the calibration and
characterization studies are not constant. In order to obtain stress-strain curves
based on constant strain rate, a few tests based on various levels of constant
incident strain wave were performed. Figure 31 shows the results of stress-strain
curves at constant strain rate. The shapes of the stress-strain curves are also

similar to those based on constant incident strain waves.

64

Table 5 Parameters of SHPB and LID of specimen used by
various researchers.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ngkinson bars
Owner Length (mm) LID of
Material Diameter Striker Incident Transmitter Spec‘men Ref'
(mm) Bar Bar Bar
|.M. Daniel 17-4PH 12.7 76~152 914 914 1.5
stainless [9]
steel
0. Sawas Cast 25.4 890 2500 2500 -------
N. S. Brar Acrylic [10]
Michael MIL-S- 19 ---- 1524 1524 1 [1]
Kaiser 468500
W. Chen VM350 12.7 152 2130 915 ------- [7]
Steel
C.T. Sun Hardened 12.7 100 760 550 ------- [4]
steel bars
Paul S. SAE-340 9.2 ----- 1220 1220 0.5 [2]
Follansbee maraging
steel
----- : No data found

Ref.: See the Reference section

65

 

 

  
 

 

 

 

200 )5 --~— 555“" ‘5“ “5555555 I
180 5 L
160 -
140 - ’
E 120 5
E
m 100 5
m —+—Strain rate=140 1/s
a»
:5 80 5
(I) , + Strain rate=170 1/s
60 5 *-
5: a r Strain rate=200 1/s
40 5
------- Strain rate=230 1/s
20 5
0 ; l
0 0.005 0.01 0.015

Strain

Figure 31 Stress-strain curves at constant strain rates for the
composte specimens with D=15.8 mm, L=25.4 mm.
(based on various levels of constant incident strain wave tests)

66

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS

1. Conclusions

Some conclusions concerning the calibraiton of the SHPB and the
characterizations for aluminum and thick laminated composites can be drawn as
follows:

A. Results from the calibrations for ampliﬁers and Wheatstone bridge circuits
seem to indicate that the electronic components of the SHPB perform as
accurate as expected.

B. Results from the calibration for 6061-T6 aluminum seem to be reasonable
when compared with the existing results, such as those obtained by Chen [5].
The accuracy of the SHPB is then veriﬁed.

C. Under constant strains, the stresses in the stress-strain curves rise as the
strain rates rise in all cases.

D. The dimensions and dimensional ratios of specimens seem to affect the
stress-strain relations signiﬁcantly.

E. There is a noticeable difference between specimens prepared from the
surface of thick laminated composites and those from the center of thick
laminated composites, implying the nonuniformity of the thick laminated
composites.

F. The difference due to material differences from the same composite is not

as signiﬁcant as that due to geometrical difference.

67

2. Recommendations

A wave-shaping technique may be required for future studies based on the
following discussion.

A. A Split Hopkinson Pressure Bar (SHPB) is a material testing device.
Instead of applying quasi-static forces to specimens, an SHPB can apply
dynamic waves to specimens for characterizations of dynamic properties. When
the impact between the striker bar and the incident bar takes place, a constant
strain wave will be generated on the impact surface. Although high strain rates
are usually desired in dynamic material characterizations, they may cause
damage or yielding to the part of specimen close to the impact surface and result
in nonuniform deformation in the specimen and violate the fundamental
assumption of SHPB. Hence, a wave-shaping technique to provide non-constant
strain waves with a moderate rise in the beginning of the wavepattern is desired.

B. It is true that a high rise in the incident strain gives a high strain rate.
However, the incident strain is only an input wave. What really occurs in the
specimen is dependent on the material properties and can be represented by the
reﬂection wave. In this thesis study, nearly-constant strain waves were used in
the tests. However, the strain rates during the tests did not remain constant.
Although a postprocessing technique may be used to convert the stress-strain
curves for non-constant strain rates into those for constant strain rates, the

microscopic behavior of the materials under non-constant strain rates is believed

68

to be different from that under constant strain rate. Hence, a wave-shaping
technique may be desired.

Since the specimen dimensions and dimensional ratios have great effects on
the stress-strain curves, a more comprehensive study on ideal specimen

dimensions and dimensional ratios should be performed.

69

APPENDICES

7O

Engineering stress (MPa)

APPENDIX A

OTHER FIGURES AND WAVE SIGNALS

 

 

 

 

350
6! H "1111919“: i
We-.. 55“)
300 ~ 16g”? MMAAAMAM Q AA
. ,{ui’dﬂ ’ MAM C}: A
‘ ‘ ; 9’"! AIWA L? AA
.2' ' A" AA 0
i I .14.]: c “KENS-“AM Q A
250 J f/IT‘;-1MA “AN-A i A
i 3"" D11.98L13.08—1901Is Q: g
g _ 5 —a—D12.06L19.16-1301/s , A
200 - ;._g_..e “f I —-——-e—D12.16L25.66-1101/s 5 .
:' .. ' o --— D12.16L6.26—1300 1Is I A
‘ D12.44L19.12-3301/s I R
. ------- D12.6L25.38-2901/s .
150 — A I ., ) 2
.5 . ~~ ~ D12.8L13.3-5401Is g
, ; D15.88L12.86-4201/s 5
g <5 --D15.8L12.86-1601Is
100 g; -—--D15.8L16.06-1101Is
"2 ' 5 -- ~95 —- D15.8L31.96-70 1/s
- - .. 08.08L8.04-1050 1/s
50 ————— D15.8L12.86-s-150 1/s
g) ..i’ .. 5D15.8L12.88-s-1601/s
‘ -.-.;_. A?
L U l 1 T l
-0.01 0.01 0.03 0.05 0.07

Engineering strain

Figure A-1 Other stress-strain curves of aluminum 6061-T6
with different strain rates.

71

Stress (MPa)

 

 

 

249 )
199 i
149 -« '
gm?!” ’
99 J —D15.8L25.56
9.43 .‘ ...... -- D15.8L25.06
fan/45’ - _ D15.8L11.33-upper
49 j [if], . - D15.8L12.73-middle
L 35-: + D15.8L12,.36-bottom
-0.001 0.004 0.009 0.014 0.019 0.024
Strain

Figure A-2 Stress-strain curves of glass-epoxy composite
specimens cutting from the same plate.

72

299 5
fine??? ’7
o. '5 . :: ‘
a '9‘“
d)
.2
a, -—D15.8L25.56
_.e_ D15.8L25.06
.9 ...- D15.8L24.64 (two 1232 together)
..._ 015.81.12.46 (two L623 together)

 

—-— D15.8L8.94 (four L2.23 together)

I I

 

T I

0.014 0.019 0.024 0.029
Strain

I

4
.Ir'— i

-0.001 0.004 0.009

Figure A-3 Stress-strain curves of assembled glass-epoxy
composite with different layers.

73

Stress (MPa)

349 5

  
  
 

 

 

299 5
249 “
199 1
14
— D15.8L25.56
99 + D15.8L25.06
.... D15.8L12.7
.... D15.8L12.4
...— D15.8L6.44
49 —.— D15.8L2.27
.4. D15.8L2.26
—1 ’2. " ‘ 1 1 T
~0.001 0.009 0.019 0.029

Strain

Figure A-4 Stress-strain curves of glass-epoxy
composite with different UD ratios.

74

 

The wave speed in the bars are calculated by:

04%

E : the length of the bar.
T: the period of the waves, calculated by the way showed below.

 

 

 

 
 
    

J VIIIUIIIHCOCII Scope 1 HI ‘il I} In: I'CI

I? £9 _ligNJQg-IDJQL ' ' .‘ . -- ' " '
Vols/Div Ref 2 50 mV

Pomon' -55rnV

SIW 751 uSocldv

 

Striking bar to incident bar at
,. ,7 the gas pressure of 175 kPa

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g:

 

 

. i L . . 7' «W:
. ., ,, 7 _ _ 3.7 g __ -. L ‘ - ,g-j» 5' “...;
L 7 ”if“ .7
L ) 3 _. 1‘11
I A A A A ' ’f‘. "25.. L‘
: ' ;‘ Xvi-IL
. 55,910
L ' . ' 7
1 v"\ r. .
1 L ‘ 3 .1, - “ :. ‘h‘lr'zél
— I , , , - - ---—--- —----——-—- —-----— - - — ..- v - - 4 . - 7,41.

 

 

 

 

 

 

 

 

 

 

l L L ‘7 5 ‘51 4’ I.
L ) , ) ‘L ‘3‘ j
’ L
) 5* 5 if *5 5 20 T 5 —n —5---5-‘~5)—----—5 ”T1"
4 $
A — ) ) ' 5]
L L L I31)
I hf":
" 1.3.1
L L ‘ 1 1 “.37 i
782m: 85.7fm

 

.1 as
gee-'2] Sid-ml. .s-‘itﬁl -. ”Eel — *

ISMIgiiﬂﬂﬁl FLIIIVMMSm-m mom-mm? ‘ ’ - ell!

 

 

Figure A-5 Calculation of wave speed in bars.

75

20

 

Reﬂected pulse

Transmitted pulse

 

Voltage (mV)

 

Incident pulse

 

 

 

30 I T T Y T I I I l I l I r I
400 o 100 200 300 400 500 600 700
Time (us)

Incident pulse. reﬂected pulse and transmitted pulse for Aluminum com—To.

 

 

 

 

 

 

 

 

 

t0
Front—end
—O—— Back-end
30-ﬁ
20-—
tO-e
0 q
'10 I 1 I I I I I
0 100 200 300 400

('t’impnrison of force at front-end and hack-end of Aluminum 606 I -'l‘(i specimen.

Figure A-6 Wave shapes and forces comparison of Aluminum
6061-Y6 with shaping technique. (from Dr. W. Chen [5]).

76

Volts/Div: Ch 0: SOmV Ch 1: 50 mV

 

 

 

 

 

 

 

 

 

Position: -5 mV -200 mV
1 i
1 . _ “.--..-
l
f ..
i ﬁx!
”___...
1‘; k‘.
1 .- , “k .
. M- 1
ivmhwvwwuhwuzww~wf *——" ““"“+.‘—_ﬂﬁtmmﬁ_*_——
1 . ‘.
¥ ‘
-_- i; -- _—LM-_..L--..---._.-;..--M___
1 1 , -
1 ‘ . “Kerk”.

 

' 53.6 ms “

Figure A-7 lncident, reﬂected and transmitted waves for the aluminum
6061-T6 specimen with 015.88 L12.86.

77

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 0 V -200 mV

   
  

 

 

 

 

 

 

 

 

 

 

 

53.9 ms

Figure A-8 lncident, reﬂected and transmitted waves for the aluminum
6061—T6 specimen with 012.8 L138.

78

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: -5 mV -200 mV

 

 

 

 

 

 

 

 

 

 

Figure A-9 Incident, reﬂected and transmitted waves for the aluminum
6061-T6 specimen with D10.68 L10.4.

79

 

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 0 V -200 mV

 

 

 

 

 

 

 

Figure A-10 lncident, reﬂected and transmitted waves for the aluminum
6061-T6 specimen with 012.44 L19.12.

80

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 10 mV -190 mV

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure A-11 lncident, reﬂected and transmitted waves for the aluminum
6061-T6 specimen with 012.6 L25.38.

81

 

 

 

 

 

 

 

 

     

 

 

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: -5 mV -200 mV . II
._ I
a;
I
. 1 U
a A_ m 4 - s- a- -___+'_s_ 1‘- 4 ___.ﬁ en“
; 1 1
i L. 1 '
i - .
L L
... ..1 W w A y _k m
. i 1 1
. - .; s _ - .4, i -_ _ M.
: . V
1 L J-‘L’MWWW " M
Di L L I? L ”Run...“
#4.)...;.:...- -.kv,_x.;—;.W_:=pr,—.xa:x.»m-.--,-,.__ ,_ ‘ .7- FN._,..‘._,,,,‘,_.'_W _ $1 , . i, .- . ..

 

 

 

Figure A—12 lncident, reﬂected and transmitted waves for the aluminum
6061-T6 specimen with 08.08 L804.

82

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: -10 mV -85 mV

 
 
   

 

 

 

 

 

 

 

 

1 l
L l

. '- v
pg.r~,..p.; .A aim“ L'IM'.V'JM—¢VMMW.~LAW" w my: Maw-
: _ 1

 

'3’:

 

 

 

 

 

 

 

55.4 ms
?"‘='.A-‘s-L'r It. '2‘"; ‘"‘-:— ‘i'm'jf’l" . -- " ~71 r * Pr ”:3". F2‘ -.i'¢"’J'.-¥f}";"?"' -- r‘rw' ‘ we:
gasps-Efﬁa'lwimamay»; uni-cw «311:1 . ., z. ~. . .

Figure A-13 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L12.4.

83

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 20 mV -50 mV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure A-14 incident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L12.96-edge.

84

Ch 1: 50 mV
—45 mV

Volts/Div: Ch 0: 50mV
Position:

35 mV

 

 

 

 

 

 

 

 

 

 

 

 

60.1 ms 1 ~ We-

5"" i,‘J‘v’,"f‘.»“.;"‘ . .‘ _ -_V
J 1“, 5 at [(13.93]; '[m ._ “ftéigt a.

 

 

 

 

l
' 60.5 ms

1" 11.-:2: {Jea_,=.t..¢§aavgz , vim , ”*“M . :45

Figure A-15 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L12.94-edge.

85

IIIiII

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 35 mV -35 mV

 

 

 

 

 

 

 

   

1 ~"
, 1 in"
__ --.-m -.__.._ _____-_____ ___ -..___2.-__.___ “Few“. --.....___-.- . :_ .14- __
ru‘ tiw' .HMWhCMp 11' yr. .1.‘ m M.”LW\AMMM‘XI MNm‘wiWﬂTywrbw‘hTIHJnﬂ-v MW-" If
1
1

 

 

1
l
i
I
1
1
i

l
1 l
I i
l

 

Figure A-16 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 012.3 L12.76.

86

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 35 mV -35 mV

 

_____.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.. ‘39? ”wt ‘ .905» - , a':x‘;'-" M»: ‘ 1 w-a . .' "1‘.
13".!" f V \'" *1? ‘lf'lVfi '. ’11:": f 11"}!7 ' ‘ 91;. "4‘ ':f;~$tﬁf”~'. ' ‘- ' “"e .

Figure A-17 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 010.84 L12.14.

87

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 35 mV -35 mV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure A-18 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 010.84 L11.28.

88

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 30 mV -60 mV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure A-19 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L25.6.

89

 

lllill

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 35 mV -60 mV

 

f

 

 

 

 

 

 

 

 

 

 

 

 

imam-an wan-um»)?! ..Wﬁ-tr'Ww-fﬁw’MAw-v- W'WLM-MWQLFNH r“! ‘
1

 

’ 1 1 3 L

 

 

 

 

 

 

Figure A-20 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L25.54.

9O

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: -10 mV -85 mV

 

 

 

 

 

 

 

. A
i , - i
s_ ___. .__ L L 1. AL ‘m
1

 

 

i

 

l .
| 1

55.9 ms
"‘3’;- '. ‘3? 1377-. “3' f‘TTFEE‘FE'ﬂE‘TT 2.54;" emit»: _ Janene; 39"? *
'7 ’ x ....la'v 4.1.? v “ “yr in-» '7" 5,‘7-.. 2". _ _ n' .115. o- r 1,. - .

l
a;
.1
#1
,n
I?
I

Figure A-21 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L127.

91

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: —10 mV -85 mV

 

 

 

 

 

 

 

 

 

 

 

 

Figure A-22 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L25.56.

92

Volts/Div: Ch 0: SOmV Ch 1: 50 mV
Position: 35 mV -60 mV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure A-23 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L12.55x2.

93

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV -60 mV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i L / 1 L
L i
WM#JM.:W.T&r-‘aﬁ.bfkg'umtha.”kwﬁvwiw¢g‘+sﬁ)ﬁzﬂfuwwMMW‘VMLﬁ}: - _ . .
1 i .L l
1 L L L
i L L l
1
52 9 ms -______._ n" “F ~' ~ “ 53.3 ms
3. $95: V 3‘. 1377‘” {iv-WW. t.“ ”1‘” at}; "M' 'F 49 6 ~$21W3 it

Figure A-24 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L12.42x2.

94

 

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV -60 mV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

53.3 ms ‘”"”—"""“”"'“““’_'“”—"”“l‘*”” I K 1 353.7ms “

v

... ‘-_4'.."_ "1‘ 3‘-, Wf-‘vvv'ﬁ?’ T . 'r ”5’ : 'H V' _ 1w J“ ' 3.x." ﬁr‘71_ :1 ‘1‘1 {v.vI1’-l,-v 7“"; W ‘, "‘3' -‘ 7". --I 'f‘ 11.3’ k! I
3W '3 m«-.‘“~ :W~‘>‘r-L~.ai‘:7-Dr+m 5;: ,, «riff. fm‘.éiv.x~:=..-aﬁﬁw é knit; ' «my; ‘tuw-v'" ~ ~

Figure A-25 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.7 L25.46.

95

 

I
I

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV -60 mV

 

 

 

 

 

 

 

 

 

 

 

 

 

 

53.7 ms ‘_'_‘ “‘”“““”"‘"—”‘“ “ ‘ “W“ “ 54.1 ms
sews.“ “5;?" east, i awe». '- :3 mﬁﬁ ‘2’"

Figure A-26 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.7 L25.18.

96

  
 

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV -60 mV

 

_l__-

 

 

 

 

.. --_.. ....— -- _. -,__ -...

 

 

 

 

 

 

 

 

‘ w
i I ,
#’W” if a, ,
1 J : A ‘ A” W.“ i E"

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 
 

: f”
i L/ xx‘
w“— _ __C I T I ‘ ._.__-_..‘._..,.____
mamas». am «my mannserwn‘mwsxmvmm-wuwmﬂ . WML
. I .
. W.:W7
a . I
50.7 ms ‘ W” 51.1ms “

Figure A—27 incident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L12.5-middle.

97

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV -60 mV

     

 

 

. '
i

 

 

 

 

 

 

 

 

i

 

 

 

 

L‘ “-'_ h __"n — k'ﬂ‘" 7"_ -VV-Hﬁ—7--‘fi_-g 7 .‘h 1% V ‘L‘_"’""‘!‘I“—”1r—“— 4' ---“..- %
I f
I /.I
I i r ' {
”Mtg-ﬂ“ MUM-w‘Frl—mmww’n'r,‘ WWr'i‘VW"~”MWW*JWW#
T . | I

 

I

     
 

 

 

 

i

i

 

 

Figure A-28 Incident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L11.44-edge.

98

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV -60 mV

   
  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

51.1 ms ‘i'
“Mkmﬁsk'lzf ‘ '

Figure A-29 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with 015.8 L11.9-edge.

99

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV -60 mV

 

 

 

-.. . ___“-..V ...

 

\‘i

 

 

 

 

 

‘ J
‘4' My“ Winn-M JMAMulp-rmqt M W M‘M‘a.ﬂ.l.wmi,A-. u» uwmwmw-‘mwn 5’“

 

 

 

 

 

Figure A-30 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with D15.8 L11.4-middle.

100

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV -60 mV

        

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure A-31 Incident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with D15.8 L11.64-edge.

101

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: -10 mV -100 mV

 

 

 

 

 

 

 

 

 

 

l l i
55.9 ms 56.3 m
9 if ur- W'”""“"“"~‘f'.'"“""‘t"'ﬁ.gr :‘rfwr‘n’ -- - -- ~- . ~9- .: ,
‘ ‘ . Ian ' _.

' 34 . '.. .O 9 - .. t 3‘3".»3 . {IV-14““: “-
. . so.” “3‘ .23; ~“ at??? er r‘: 41 waif: 4 Ea; 3.9.2.: .3

Figure A-32 Incident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with D15.8 L25.06.

102

 

Volts/Div: Ch 0: 50mV Ch 1: 50 mV
Position: 35 mV 60 mV

 

 

I
——4~ v——4 _ ——+v ___—.—

 

 

 

 

 

 

 

 

 

 

 

 

 

. I
l I
' . I
I I I
J
m I a
I r" ' ‘x I I
J I -
4P}, ‘ '~ I
K- I
7 Y*— ﬁ .——— —— , iiﬁiivﬁ ~ ..—__.__ —A 4“,“ T — .9
ix" jI L's,
fr ['1‘

 

 

 

 

I
I I I

 

I
dunk 4),“ ‘1}.NJ-"O—lWM-‘MVL-nﬁW~gmwblwwﬂwﬁﬁr*‘~fl~ﬁpkf1 Wi
,_ ...... ._ _,.___¥_-- ,, .... Wim—
I
I
I

   
  

 

----.__,._~__.____._ H}. __M _ .--.1..._. ' ._. 5 53 i3 ms ___.

 

. I; ; ""_" .',r > ;, tiff"? ' _ -. . . EV, 151*" 1r : ., ' J _ my! 9.99 E” _. _ 'ﬁz‘mﬁ'.’ \ ‘.‘+..._. 'I‘ if. *4;
. 5' v.51: “"59... ‘,-.“’"'.*.r" ._‘ ‘ 5:..." " zﬁtgmu'ﬁg , ;,._«‘r$‘ sz‘it-"fc-ﬁmi‘a 1 A :::s‘.;:i.i--" “‘tﬁ‘ﬁ, r_,.~,_’,‘v‘.' "3.:

Figure A-33 lncident, reﬂected and transmitted waves for the glass-epoxy
composite specimen with D15.8 L12.6.

103

APPENDIX B

SPLIT HOPKINSON PRESSURE BAR THEORY

a. Strain and Strain Rate Equations

Z
}/ 1 l l———> y
X \ ‘
.9

dy

 

 

 

Figure A-34 Pressure bar shown with differential element.

The ﬁgure above shows a differential element with length dy and cross
sectional area A0 in a pressure bar. After impact, the following equation can

describe the motion of the pressure pulses.

2
AoE%—A0E%‘y3 = Aodypa—lﬂ (A4)

at2

where E is Young’s Modulus of the bar material, u1 and u2 are the displacements
of the differential element in both ends, ,0 is the density of the bar material.

As the wave velocity ,Co , can be calculated from

,0 (A-2)

104

the equation A-1 can be simpliﬁed for the bar’s equation of motion.

aul auz azul
Cozl— — —] = ,
5y 5y at

 

(A-3)

If the rates of change of displacement of the two sides of the element are equal,

the equation of motion can be written as

uz - u1+ alld
ay y (A-4)

Upon differentiation, equation A-4 becomes

6u2 6u1 52111
: + 2
6y 6y 6y

 

 

dy (A-5)

By substituting equation A-5 into equation A—3, the equation of motion for the bar

becomes

_C0202u1:52u1 (A-6)
8y2 6:2

 

 

For harmonic waves,

E 6 But _ 60'
"‘ — -_ (A-7)
6y 5y 6y

105

where 0' is the stress across the cross section.
So the equation of motion can be rewritten in terms of the pressure and velocity

across the bar cross section as

_60(y,t)_ Q1
ay -p6t (M3)

For a uniaxial state of stress, the pressure is equal to the stress over the
pressure bar cross section, that is .p ( y , t) = 0- ( y , t) . If we assume a positive

traveling harmonic wave of the form
P(y,t)= Pei(W’"ky) (A-9)

where P is the amplitude of the pressure, w is the frequency, tis the time, k is the

wave number and is deﬁned as k = 1. . Taking the ﬁrst derivative of

C 0
equation A-9 with respect to y,

M : -ikPei(Wt-ky)

6y (A-10)
Substituting this derivative into equation A—8,
(3 . - _
p—Z-z sze '(Wt ky) (A-11)

at

so the particle velocity can be given by equation A-12

106

v(yJ) = iPei(W"ky) = —k— 0'(y,t) = —l— o-(y,t) (A-12)
W W pCO

As 0'( y, t) = g( y, t)E , the particle velocity in terms of the bar strain can be

express as

V(y,t) = C0£(y,t) (A-13)

For a negative traveling wave the particle velocity is:

V(y,t) = -C0€(y,t) (A-14)

with above equations, specimen strain rate can be calculated easily.

 

 

 

 

 

 

 

 

' E
_—'51 ___}
8R «— I AS IA 57'

|< L ,

(IS) (28)

Figure A-35 Parameters of cylindrical specimen and bars

107

For the specimen shown in Figure A-2, The average strain at any time is

given by

1115- uzs '
85 = (A-15)

 

the average strain rate is:

6183 : Uls-Uzs (A-16)

dt L

 

 

the velocity at interface 18 is comprised of the incident ( positive traveling wave)

and the reﬂected (negative traveling wave) as
uls=C051—C0£R . (A.17)
the velocity of interface ZS is a positive quantity as
U 23 = C 0 6‘7 (A-18)

so the specimen strain rate in terms of the bar strains as

dé's : _ C0(8T - 6‘1 + SR) (A-19)
dt L

 

 

108

If the specimen deforms uniformly, such that

6'1 + 8R = 6T (A-20)

the equation for the specimen strain rate can be reduced to

dé‘s _ _ 2C0 5‘
dt L (A-21)

 

 

which can be integrated to get the specimen strain

I
I6 R (t)dt (A-22)
O

2C0

 

85(1) =—

b. Stress Equation
The average force on the specimen is given by

P1s + st

P v = _
a 2 (A 23)

 

the forces at the ends of the specimen can be expressed in terms of the incident

and reﬂected bar strains as

P15 = EA (291+ 51:) (A-24)

109

P25: EA £1

(A-25)

So the average force on the specimen in terms of the bar strains can be given as

Pav = iii-(£1 + (9R + 8T)

with the equation A-20, and the equation followed,

Pav
As

 

Us:

the expression for the average specimen stress is:

 

0's(t)=E: 61(t)

S

110

(A-26)

(A-27)

(A-28)

REFERENCES

111

REFERENCES

[1] Michael Adam Kaiser, Advancements in the split Hopkinson bar test, Master’s
thesis, Blacksburg, Virginia, 1998.

[2] Paul S. Follansbee, Los Alamos National Laboratory. “High strain rate
testing—the Hopkinson bar”.

[3] H. Zhao, G. Gary and J. R. Klepaczko, “On the use of a viscoelastic split
Hopkinson pressure bar”, International Journal of Impact Engineering, 19 (4),
319-330, 1997.

[4] Lal Ninan, J. Tsai and CT. Sun, “Use of split Hopkinsin pressure bar for
testing off-axis composites”, International Journal of Impact Engineering, 25, 291-
313, 2001.

[5] W. Chen, ‘the testing result of 6061-T6 aluminum with shaping technique’,
sent by email, 2002.

[6] D. J. Frew, M. J. Forrestal andW. Chen. “Pulse shaping techniques for testing
brittle materials with a split Hopkinson pressure bar”, Experimental Mechanics,
42(1), 93- 106, 2002.

[7] D. J. Frew, M. J. Forrestal and W. Chen, “A split Hopkinson pressure bar
technique to determine compressive stress-strain data for rock materials”,
Experimental Mechanics, 41 (1 ), 44-46, 2001.

[8] W. Chen and F. Lu, “Dynamic compression testing of soft materials”, ASME
Journal of Applied Mechanics, 69, 214-223, 2002.

[9] H. M. Hsiao, I. M. Daniel and R. D. Cordes, “Dynamic compressive behavior
of thick composite materials, Experimental Mechanics, 38, 172-180, 1998.

[10] O. Sawas, N. S. Brar and R. A. Brockman, “Dynamic characterization of
compliant materials using an All-polymeric split Hopkinson bar”, Experimental
Mechanics, 38, 204-210, 1998.

[11] Christophe Bacon and Arnaud Brun, “Methodology for a Hopkinson test with

a non-uniform viscoelastic bar”, International Journal of Impact Engineering, 24,
219-230, 2000.

112

 

[12] Frank E. Hauser, “Techniques for measuring stress-strain relations at high
strain rates”, Experimental Mechanics, 395-402, 1966.

[13] J. F. Bell, “An experimental diffraction grating study of the quasi-static
hypothesis of the split Hopkinson bar experiment”, J. Mech. Phys. Solids, 14,
309-327,1966.

[14] Robert L. Sierakowski and Shive K. Chaturvedi, “Dynamic loading and
characterization of ﬁber-reinforced composites”, John wiley&sons, inc. New York,
41-77, 1997.

[15] Jonas A. Zukas, Theodore Nicholas and Hallock. F. Swift, Impact Dynamics,
John Wiley & sons Inc. 287-308, 1982.

[16] Karl F. Graff, Wave Motion in Elastic Solids, Ohio State University, 75-134,
1975.

[17] D. R. Veazie, S. W. Park and Min Zhou,”Post-impact behavior of polymeric
composites and the effects of salt water aging on tensile properties”, 1999.

[18] Christophe Bacon, “Separation of waves propagation in an elastic or
viscoelastic Hopkinson pressure bar with three-dimensional effects”, International
Journal of Impact Engineering, 22, 55-69, 1999.

[19] David E. Lambert and C. Allen Ross, “Strain rate effects on dynamic fracture
and strength”, International Journal of Impact Engineering, 24, 985-998, 2000.

[20] C. Bacon, “Numerical prediction of the propagation of elastic waves in

longitudinally impacted rods: application to Hopkinson testing”, International
Journal of Impact Engineering, 13 (4), 527-539, 1993.

113

 

   

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