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Heterogeneous shock energy deposition in shock wave consolidation of metal powders
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Mutz, Andrew Howard
(1991)
Heterogeneous shock energy deposition in shock wave consolidation of metal powders.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/makt-d357.
Abstract
Shock wave consolidation of powder is a high deformation rate process in which a shock wave generated by an explosive or a colliding projectile rapidly densifies and bonds together the powder particles into a solid compact. The deposition of the shock energy during this process is highly inhomogeneous on the powder particle scale. Evidence of the extent and pattern of the energy deposition was provided by recovery experiments performed using an initially crystalline alloy which solidifies to a metallic glass upon rapid quenching from the liquid state. The amount of metallic glass was measured and analyzed using a heat flow model. The energy deposited during the shock wave passage was best modeled as deposited partly into the particle bulk and partly onto particle surfaces. To investigate this inhomogeneity, and the powder parameters which influence it, a propellant driven gas gun was designed, built and utilized. The planarity of the shock waves produced using the targets designed for the gun was established. Powder-powder thermocouples were impacted with powders of varying sizes to establish the effect of particle size on energy deposition. Small particles in contact with large ones were inferred to absorb the greater fraction of shock energy. Hardened and unhardened steel powder was shocked to investigate the effect of particle hardness on energy distribution. The recovered compacts were not measurably affected by the initial hardness. Compaction experiments were performed on a Ni based super-alloy and on a SiC reinforced Ti matrix composite to test some of the practical applications of the process and the target designs developed. Superior tensile properties were observed in the shock consolidated and heat treated Ni based 718 alloy. The SiC reinforced composite was recovered in the intended net shape with no macro-cracks in the compact body, but with fractured SiC particles.
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California Institute of Technology
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Engineering and Applied Science
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Applied Physics
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Vreeland, Thad
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23 May 1991
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CaltechETD:etd-06282007-091349
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HETEROGENEOUS SHOCK ENERGY DEPOSITION IN
SHOCK WAVE CONSOLIDATION OF METAL POWDERS

Thesis by

Andrew Howard Mutz

In Partial Fulfillment of the Requirements
for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

1991
(Submitted May 23, 1991)

TO MY PARENTS

—iii—

ACKNOWLEDGMENTS

Knowing full well there are unintended omissions, I gratefully thank Prof. Thad
Vreeland for his teaching, advice and encouragement, and both Prof. and Mrs.
Vreeland for their friendship and hospitality.

I surely would not have been able to work at Caltech for so long without the
entertaining distractions which made life here pleasurable. Barry Krueger provided
many of those distractions, as well as able scientific help completing the Keck
Dynamic Compactor, and his untimely death leaves a deep silence. The help and
humor of Concetto Geremia are also unforgotten in his absence. Dave Lee sparked
nearly electric excitement at times. I also thank Phil Askenazy, Pascal Yvon,
Brigitte Kruse, Doug Pearson, the Keck lab applied physics and material science
folks, George Clovis, the Caltech fencing team, and the Garveys and Infra—Red Sox
softball teams.

Technical collaborations are also numerous. Dr. Prakash Kasiraj performed the first
of the shock consolidation experiments on Markomet 1064 on the 20mm gun in the
Caltech Seismology Lab, in collaboration with Prof. Thomas Ahrens. Prakash also
taught me the fundamentals of shock physics. Prof. Naresh Thadhani made the
TEM and optical observations of Markomet 1064 and Pyromet 718. Dr. Ricardo
Schwartz collaborated on the design and construction of the thermocouple
experiments. The members of Caltech’s Central Engineering Services machine shop,
led by Norm Keidel and Louis Johnson, not only fabricated most of the Keck

Compactor but also suggested design changes which caused it to work.

-~iy—

ABSTRACT

Shock wave consolidation of powder is a high deformation rate process in
which a shock wave generated by an explosive or a colliding projectile rapidly
densifies and bonds together the powder particles into a solid compact. The
deposition of the shock energy during this process is highly inhomogeneous on the
powder particle scale. Evidence of the extent and pattern of the energy deposition
was provided by recovery experiments performed using an initially crystalline alloy
which solidifies to a metallic glass upon rapid quenching from the liquid state. The
amount of metallic glass was measured and analyzed using a heat flow model. The
energy deposited during the shock wave passage was best modeled as deposited
partly into the particle bulk and partly onto particle surfaces. To investigate this
inhomogeneity, and the powder parameters which influence it, a propellant driven
gas gun was designed, built and utilized. The planarity of the shock waves produced
using the targets designed for the gun was established. Powder—powder
thermocouples were impacted with powders of varying sizes to establish the effect of
particle size on energy deposition. Small particles in contact with large ones were
inferred to absorb the greater fraction of shock energy. Hardened and unhardened
steel powder was shocked to investigate the effect of particle hardness on energy
distribution. The recovered compacts were not measurably affected by the initial
hardness. Compaction experiments were performed on a Ni based super—alloy and
on a SiC reinforced Ti matrix composite to test some of the practical applications of
the process and the target designs developed. Superior tensile properties were
observed in the shock consolidated and heat treated Ni based 718 alloy. The SiC
reinforced composite was recovered in the intended net shape with no macro—cracks

in the compact body, but with fractured SiC particles.

—V—

TABLE OF CONTENTS

Acknowledgements

Abstract

List of Illustrations

List of Tables

Chapter 1. Introduction to shock consolidation.
1.1 Overview.
1.2 Historical background.
1.3 Continuum shock waves in porous media.
1.4 Particle scale shock waves in porous media.
1.5 Scope of this work.
References
Chapter 2. The Keck Dynamic Compactor.
2.1 Introduction.
2.2 Design.
2.3. Wave Planarity Experiment.
2.4 Analysis.
2.5 Conclusions.
References
Chapter 3. Melt fraction and energy distribution during

3.1

shock consolidation.

Introduction.

iii
iv
viii

xii

12
12
13
15
16
18
27

Chapter 4.

Chapter 5.

Chapter 6.

3.2 Markomet 1064 experiments.

3.3. Discussion.

3.4 Energy deposition modeling

References
Size distribution and energy deposition:
Powder thermocouple experiments.

4.1 Introduction.

4.2 Thermocouple experiment design.

4.3 Experimental results.

4.4 Discussion and analysis.

4.5 Conclusions.

References
Hardness and deformation in shock
consolidation.

5.1 Introduction.

5.2 Experiment design.

5.3 Results and discussion.

5.4 Conclusions.

References
Tensile properties of a shock consolidated Ni
superalloy.

6.1 Introduction.

6.2 Experiment.

30
32
33
44

46
48
51
52
58
74

76
76
17
79
87

88
89

—vii-—

6.3 Results and discussion.
6.4 Conclusions.
References
Chapter 7. Multiple cavity and near net shape shock
consolidation.
7.1 Introduction.
7.2 Experiments.
7.3 Discussion.
7.4 Conclusions.
References
Chapter 8. Conclusions and remaining issues.
References
Appendix A. HUGONIOT, a program for solving the shock
conditions in a powder using the Simons and Legner model.
Appendix B. THERCO, a program for simulating thermal

deposition and relaxation in a powder thermocouple junction.

90
93
101

102

103

104

106

111

112
114

115

126

—vili—

LIST OF ILLUSTRATIONS

Figure

1.1 Micrograph of polished and etched shock consolidated
spherical nickel powder.

2.1 Cutaway illustration of the Keck Dynamic Compactor.

2.2 Photograph of the open breech with the breech sleeve
attached.

2.3 Photograph of the compactor muzzle and extension.

2.4 Photograph of the recovery tank interior.

2.5 Cross section illustration of the target ring.

2.6 Photograph of the polished and etched Metglas
compact.

2.7 The one dimensional plane wave time — displacement
history calculated for the Metglas experiment.

3.1 Photograph of the polished and etched Markomet 1064
powder, after annealing.

3.2 Illustration of the target assembly used in Markomet
1064 shock consolidation experiments.

3.3. Photograph of the polished and etched Markomet 1064
compact, shot #814.

3.4 Plot of calculated maximum, calculated minimum, and
measured melt fraction vs. shock energy in the Markomet

experiments.

20
21

3.5

3.6

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12
4.13

Plot of temperature vs. cubed radius at the end of
shock rise time calculated with four different energy
deposition profiles.

Plot of predicted and experimental melt fraction vs.
energy, using four energy deposition profiles.
Photographs of the copper and constantan powders.
lustration of the thermocouple target.

Photograph of the assembled thermocouple target.
Plot of EMF vs. time, from thermocouple shot #77,
and simulated EMF history from model.

Plot of EMF vs. temperature for a copper—

constantan thermocouple with a 20°C reference temperature.

Plot of EMF vs. time from thermocouple shot #80.
and simulated EMF history from model.

Plot of EMF vs. time from thermocouple shot #91.
and simulated EMF history from model.

Plot of EMF vs. time from thermocouple shot #106.
and simulated EMF history from model.

Plot of EMF vs. time from thermocouple shot #108.
and simulated EMF history from model.

Plot of EMF vs. time from thermocouple shot #109.
and simulated EMF history from model.

CCD camera image of EMF vs. time from shot #109.
Schematic representations of thermal model.

Plot of energy flux vs. position during the shock

rise time, under varying amounts of energy bias.

72
73

5.1

5.2
5.3

5.4

5.5

5.6

6.1

6.2
6.3

6.4

6.5

6.6

6.7

7.1

Scanning electron micrograph of M350 maraging steel
powder.

Photograph of porous bronze target insert.
Micrograph of polished and etched consolidated steel
powder, heat treated before shock.

Micrograph of polished and etched consolidated steel
powder from near edge of large compact.

Micrograph of polished and etched consolidated
mixture of hardened and unhardened maraging steel.

Scanning electron micrograph of fracture surface of

shock consolidated tensile specimen of hardened maraging steel.

Scanning electron and optical micrographs of
Pyromet 718 powder particles, as received.
Illustration of tensile sample and frame.

Optical micrographs of shock consolidated and hot
isostatically pressed samples of Pyromet 718.

Plots of yield strength and ultimate tensile

strength vs. aging time for shocked and hipped Pyromet 718.
Bright field TEM micrograph of an interparticle
region of shock consolidated Pyromet 718.

Bright field TEM micrographs of an intraparticle
region of shock consolidated Pyromet 718.

Bright field TEM micrograph of shear bands in shock
consolidated Pyromet 718.

Nllustration of a four cavity porous bronze target

insert.

82
83

94
95

100

107

7.2

7.3

7.4

Micrograph of polished and etched shock
consolidated 5.5—6.5 um spiked spherical Ni.
Photograph of recovered M350 maraging steel sectors
of shot #63.

Photographs of the green compact of Ti + SiC and

the shock consolidated compact.

108

109

110

Table
2.1

3.1

3.2

4.1

4.2

4.3

5.1

5.2

—xii-

LIST OF TABLES

Parameters used to calculate shock conditions in
Metglas MBF 50 experiment.

Shock conditions and quenched melt measurements of
Markomet 1064 consolidation experiments.
Thermodynamic and physical parameters for Markomet
1064 alloy.

Shock conditions and results of the copper —
constantan thermocouple shock experiments.

Copper and constantan thermodynamic data.

Model parameters for thermocouple simulations.
Parameters in M350 maraging steel shock
consolidation experiments.

Heat treatments and resulting tensile properties of

shock consolidated M350 maraging steel.

CHAPTER 1
INTRODUCTION

1.1 Overview

Shock wave consolidation is a process for forming low porosity, well bonded
solid material from an initially very porous powdered material. A shock wave,
generated by a high explosive or high velocity impactor, is driven through the
powder. The shock wave densifies and bonds the media of interest. Ideally, a
completely nonporous and fully bonded solid results. Interest in the process stems
from the promise of forming parts of materials not amenable to conventional casting
or sintering processes. For example, diamond powder has been explosively
compacted to a dense, polycrystalline solid {1,2]. Successful static sintering of
diamond requires treatment at approximately 2000° K and 60 kbar [3], and is
difficult though commercially viable. Also, metastable materials such as amorphous
metal alloys may only be synthesized in powder, ribbon, or sheet form. These
materials can be consolidated to bulk solids via shock wave consolidation [4].

Several difficulties with shock consolidation exist. Reflected tensile waves
generated by the interaction of the shock wave and the target assembly often act to
disassemble the shocked material. The powders do not always completely bond to
one another. The compacts produced may have very large residual strains. These
difficulties are magnified with very hard, brittle materials, which are often of
greatest interest. Research on improved target geometries [5], particle scale shock

wave models [6], and the use of post—consolidation annealling [7,8] are addressing

~Q—

these problems, but the ultimate commercial success of shock wave consolidation as
a manufacturing technique is uncertain [9].

The shock consolidation process functions by preferentially depositing energy
at particle junctions, bonding the powder together while leaving particle interiors
relatively cool. In this manner, total energy well below that required to melt the
material is sufficient to densify and bond the powder to solid. This preferential
heating is often intense enough to melt portions of the powder particles.

In this thesis, the preferential deposition of shock energy is experimentally
investigated in several metallic systems. Recovered samples are examined for
evidence of melting and freezing during shock consolidation. This data is analyzed
to determine the extent of energy localization, and the dependence of total energy
and hardness on this phenomenon. Real—time evidence of preferential heating is
gathered through the use of powder thermocouples, and used to determine the effect

of particle size and size distribution on energy localization.

1.2 Historical Background

Shock wave consolidation of powders was first studied as a means of
fabricating material in the late 1940s and early 1950s. Powders were placed in
waterproof bags and submerged in a pressure vessel. A gunpowder charge
generated the pressure wave in the water and compacted the powder. This method
was used to compact titanium carbide particles with nickel as a binder [10]. This
method was difficult to characterize and was supplanted by direct explosive
application [11] and by striking the powder with explosively accelerated pistons [12].
Early work built on explosive and shock wave data generated during WWII,
especially during the Manhattan project, when the first plane wave experiments

were conducted [13]. WWII and post—war secrecy no doubt lead to separate and

~3—

independent efforts in shock physics in several nations. Early shock consolidation
experiments were hindered by the lack of basic research and knowledge about the
high pressure dynamic behavior of solids and powders; the systematic studies of high
pressure behavior of materials published in the late 1950s provided a foundation for
the study of dynamic compaction. Materials were studied under moderate shock
pressures of 100—150 kbar by Walsh and Christian [14]. Higher pressures (400-4000
kbar) were used to characterize materials by Al’tschuler et al. [15]. Primed now with
physical data, systematic study of shock consolidation could begin. A combination
of successful shock techniques including explosive welding [16] and diamond synthesis
[17] sparked new interest in shock wave consolidation of powders. Explosive powder
consolidation was demonstrated by Pearson in 1961 [11], and Porembka et al.
successfully consolidated uranium dioxide for nuclear fuel in 1963 [18].

Active research in shock wave powder consolidation has continued since the
mid 1960s. Early systematic studies were carried out by Bergmann and Barrington
[7] and by Kormer et al. [19]. This effort has not resulted in widespread
commercialization of the process. While high quality samples of metal and ceramic
materials have been manufactured on a small scale [5,20], the process has either been
unsuccessful or uncompetitive on a commercial scale. Elimination of cracking and
difficulties with scale-up have proven formidable obstacles. Particle scale modeling
has made inroads towards an understanding of factors affecting interparticle
bonding [6]. Shock wave simulations at the continuum level have aided the
development of fixtures which minimize cracking [5]. Thorough reviews of the
research work done in shock consolidation have been compiled by Gourdin [21] by

Thadhani [9], and (for ceramics) by Graham and Sawaoka [22].

_4—

1.3 Continuum Shock Waves in Porous Media

The study of shock waves in powders begins with the study of shock waves in
a continuous medium. Excellent texts on shock wave phenomena by Zel’dovich and
Raizer [23] and by Kinslow [24] explain the following material at greater length. The
changes in density, velocity, and internal energy across the shock wave front are
constrained by the conservation of mass, momentum and energy. These constraints,

reformulated as the Rankine—Hugoniot relationships, are contained in the following

equations:

1.1) P—Py=po Vu

1.2) po V = p,(V —u)

1.3) E,— Ey = 5 (P + Po) (5-5)

where P is the shock pressure, Po is the initial pressure, V is the shock wave
velocity, u is the velocity of material behind the shock front, po is the initial density,
p,is the final density, E, is the specific energy of material behind the shock front,
and Ep is the specific energy of material before the shock front. The initial pressure
is often assumed as zero; the shock pressure is typically 10° bar. This trio of
equations includes four unknowns.

The missing piece of the puzzle is the pressure — material velocity
relationship for the material being shocked. This relationship, known as the
material’s ‘Hugoniot’, allows determination of all other continuum parameters of
interest. The Hugoniot of a material is typically derived by measuring the velocity

at which a shock wave (generated by an impact with a material of known shock

—5—

properties) propagates. If neither material is well parameterized, the particle and
shock wave velocity in one material are simultaneously measured.

The most significant difference (for shock wave consolidation) between shock
waves in solids and in porous media is the far greater residual increase in density in
a porous material. The work done by the shock pressure acting through the density
change is retained in the shocked material, and therefore a porous material is heated
significantly by the passage of a shock wave. For example, a 10 GPa shock wave
travelling through solid copper and then released will heat the copper by 6°C [25]. A
wave of the same pressure passing through porous copper containing 40% voids will
heat the copper by about 950°C [26].

The measurement of shock Hugoniots for each powder material and porosity
is fortunately not necessary. Many porous materials have been studied [27] (and the
results tabulated), and models such as that developed by Simons and Legner [28],
provide accurate predictions of shock Hugoniot data in powders from the solid shock
Hugoniot data. The Simons and Legner model predicts the powder Hugoniot using
the material’s density, p, isentropic compressibility, «, Gruneisen coefficient, y, and
distension, m. The distension is the solid density, p,, divided by the powder
density, Py). The Gruneisen coefficient is a dimensionless thermodynamic parameter

defined as

_ aR
1.4) b= ap:

The particle — velocity, u, vs. shock pressure, P, relationship given by this model is:

weak mil +P «)-1

1.5) .
Po 1+ (1 + p/2)P «

—~§—

The Gruneisen coefficient can be calculated, but the value is often derived by fitting
powder Hugoniot measurements to the model. Typical values range between 1.2
and 2.5.

Shock rise time is the final parameter of interest in the continuum Hugoniot.
How quickly and over what distance do material properties change from pre—shock
to shock values? In solids, the shock front may be as thin as a few atomic layers
[29]. In a powder, the particle size limits the sharpness of the discontinuity. This
follows logically; as the material ceases to appear continuous, the shock wave front
cannot further sharpen. Schwarz et al. measured the rise time of a shock wave to be
of the order of the traversal time of a single particle by the shock wave [30]. The
2—D simulations of Berry and Williamson predict similar rise times [6]. Some

additional data on this subject is presented in Chapter 4.
1.4 Shock Wave Behavior at the Particle Scale

Unlike the continuum behavior of shock waves, the particle scale propagation
of shock waves is not very well understood. The difficulty stems from the rapidity
of the process, the variations in material behavior, and the variations in particle
morphology. For particles in the 100ym range, all deformation will typically take
place in 50 ns, the time for a shock wave travelling at 2 km/s to traverse the
particle diameter. (See chapter 4 for experimental verification.) During this time,
parts of the particle will experience large (on the order of 100%) strain. Strain rate
effects in the 10") s range are not easily measured; dynamic Kolsky bar experiments
typically reach strain rates of 10° /s [31]. Beyond this, dynamic stress—strain
measurements are very difficult, and therefore constitutive relationships for

materials under shock deformation are typically extrapolated from lower strain rate

data.

Ballistic punch experiments can achieve strain rates similar to those in shock
consolidation; shear strains of 100 were observed in 2024—T6 aluminum struck by a
flat—ended projectile, and the shear bands formed in less than 12 ps [32]. Strain
localization occurs by the formation of shear bands. Large shear stresses occur in
shock wave consolidation of powders, but it is unclear whether shear band formation
is a dominant mode of strain localization for most metals during shock
consolidation.

Above some shear strain rate, most metals will exhibit shear instability and
strain localization as plastic deformation causes thermal softening to exceed strain
hardening [33]. Very narrow ‘microshear bands’ were observed in particle interiors of
a shock consolidated nickel—based alloy [34].

More generally, the localization of strain near particle surfaces is a feature of
shock wave consolidation of powders. The problem has been numerically modeled
for shock wave consolidation in two dimensions (rod consolidation) [6]. Under
shock conditions the momentum of accelerated material is a major perturbation to
the quasistatic case. Average and maximum strains increase dramatically, and the
characteristic scalloped particle boundary pattern is observed in the plane parallel
to the shock propagation direction, as shown in Fig. 1.1. In a series of consolidation
experiments at different pressures and energies with spherical titanium alloy powder
[13], the transition from densification without bonding to complete bonding was
explored. Onset of bonding was first observed at particle junctions with an angle of
nearly 45° to the shock direction. This corresponds to the direction of maximum
shear for the impact of one sphere into the other. As energy was increased, the
bonding became uniform.

Strain energy localization is often sufficient to melt material near particle

boundaries [35,36]. Melting can aid in interparticle bonding by dispersing surface
contaminants and welding particles together. Some systems will only bond well if
some interparticle melting is present [37]. The presence of melted material may be
inferred from the etching behavior of melted and rapidly quenched material. This
technique was used to evaluate the fraction of material melted as discussed in
Chapter 3. The degree of strain localization is expected to depend on a wide variety
of parameters including material strength, thermal softening rate, thermal
conductivity, strain hardening behavior, particle size and shape distribution, and
the macroscopic shock parameters. Some of these parameters are explored in this

thesis, but the phenomena is not completely understood.

1.4 Scope of this work

Particle scale energy deposition by shock waves is the subject addressed
herein. The influences of particle size and shape are investigated within material
systems, and some hardness effects are also explored. Both recovery experiments
and real-time data are examined. Some of the ideas developed are applied to the
fabrication of a near—net shape ceramic reinforced metal matrix composite. In order
to conduct these experiments, a 35mm smooth—bore propellant driven gas gun was

constructed. This facility is described in some detail.

Fig. 1.1 Micrograph of polished and etched shock consolidated nickel. The

powder particles were initially 125—150ym spheres, and were shocked by a 303 SS
flyer plate travelling at 1.08 km/s. The shock wave travelled from the top to

bottom in the section shown.

—10—

References

oR we NP

10.
11.
12.
13.

14.
15.

16.
17.
18.

19.

20.
21.
22.

23.

24.

D. K. Potter and T. J. Ahrens, Appl. Phys. Let. 5, 317 (1987).
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R. H. Wentorf, R. C. DeVries, and F. P. Bundy, 208, 873 (1980).
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p. 129.

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25.

26.

27.

28.

29.

30.

31.

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33.
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35.
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37.

—~11—

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of Shock— Wave and High—Strain—Rate Phenomena, edited by L. E. Murr, K.
P. Staudhammer, and M. A. Meyers (Dekker, New York, 1986), p. 313.

(i ayy J. Duffy, and R. H. Hawley, J. Mech. Phys. Solids. 35, 283
1987).

A. L. Wingrove, Met. Trans. 4, 1829 (1973).
H. C. Rogers, Ann. Rev. Mater. Sci. 9, 283 (1979).

N. N. Thadhani, A. H. Mutz, P. Kasiraj, and T. Vreeland Jr., in Metallurgical
Applications of Shock~ Wave and High—Strain—Rate Phenomena, edited by L.
E. Murr, K. P. Staudhammer, and M. A. Meyers (Dekker, New York, 1986),
p. 247.

D. Raybould, D. G. Morris, and G. A. Cooper, J. Mat. Sci. 14, 2523 (1979).

A. M. Staver, in Shock Waves and High—Strain—Rate Phenomena in Metals,
edited by M. A. Meyers and L. E. Murr (Plenum, New York, 1981),p. 877.

P. Kasiraj, in Shock Wave Consolidation of Metallic Powder (PhD Thesis,
California Institute of Technology, 1985), p. 121.

—~12—

CHAPTER 2
THE KECK DYNAMIC COMPACTOR

2.1 Introduction

A recurring issue in the shock consolidation of powders by explosive
compaction is the pressure—energy history of the resulting compact. The highly two
or three—dimensional nature of the shock front in cylindrical and capsule type
recovery fixtures leads to a variety of shock conditions in the compact. This
variation itself changes with each new material shocked, which complicates efforts
to characterize shock conditions in an assembly by calculating the history for one
powder and extending the results to other materials [1]. The availability of
powerful computing tools is beginning to ameliorate this difficulty, but analysis is
still a difficult procedure. A one—dimensional shock condition is therefore very
desirable if one wishes to measure material properties in response to particular shock
wave pressures or velocities. This refers specifically to the initial shock front which
consolidates the powder. In spite of clever momentum traps and geometries,
reflected waves of varying magnitude and direction pass through the consolidated
specimen. In a successful consolidation experiment these waves are not strong
enough to fracture the consolidated material produced by the initial shock front.
The issue of energy history is then simplified to a Hugoniot analysis of a
one—dimensional shock condition. The pressure history is equally simple during the
passage of the first shock front. The one—dimensional analysis is, of course, valid

only in the region of material unaffected by edge waves created at the outer edges of

—13—

the target and flyer.

To exploit the advantages of the one—dimensional geometry, a propellant gun
has been built and used. The Keck Lab Dynamic Compactor facility was designed
with shock consolidation and recovery in mind. The tested velocity range is
between 600 and 1600 m/s. The design velocity of 2000 m/s has not been utilized;
most metallic materials are consolidated in the velocity range of 750 to 1300 m/s.
The 3m by 35 mm inner diameter (ID) smooth bore cannon barrel accelerates a 31.5
mm diameter flyer plate mounted on a nylon sabot 25 mm long down the evacuated
barrel. The smokeless shotgun powder (nitrocellulose) used provides a stable, safe
source of propellant gas. An electrically heated wire ignites the gunpowder. The
velocity of the projectile is measured to within 1% by timing the interval between
the interruption of two light beams. The target fixture is held to the end of the
barrel under vacuum. The barrel itself is mounted on a damped carriage, to provide
vibration isolation for the building; the compactor is installed on the third floor. A
ductile iron tube surrounds the barrel as a safety shield. See Fig. 2.1.

The breech is unusual in that the cavity containing the gunpowder during
ignition is of smaller diameter than the barrel ID. See Fig. 2.2. A maraging steel
sleeve is designed to hold the powder and the ignition wire, which is sealed from the
rest of the barrel. The gunpowder is not evacuated with the barrel. The sleeve fits
closely in the breech, and screws to the breech block, where a small hole provides
room for the ignition wire. The breech reinforcement and breech block are of C300
maraging steel and the breech reinforcements have been shrink fit into place to
provide a permanent compressive load in the breech. A safety interlocked firing
system is operated from a neighboring room.

The compactor is instrumented at the barrel extension, a six—inch tube with

a threaded collar to attach it to the barrel, as shown in Fig. 2.3. An additional one—

—j4—

inch long aluminum tube provides a sacrificial coupling to the target. This tube
permits the sabot to clear the barrel extension before the flyer impacts the powder.
The 17—4 PH steel barrel extension has four ports for the optical velicometry
system, a single vacuum connection, and a fitting for a plastic microwave radar
antenna. A doppler radar [e. g. 2] velocity system supplements the optical velocity
measurement. A standard K band (25 GHz) doppler module (Plessey model
GDHM32) was adapted to measure the projectile velocity during the entire time of
flight. Though dispersion renders the measurement difficult at the muzzle, the
radar does provide data on initial acceleration and time of flight. Flyer acceleration
is essentially complete within the first 2m of barrel length. The terminal velocity is
therefore accurately measured by the optical system. This provided a guide for
choosing an appropriate propellant, and allowed estimates of breech pressure. (The
radar is no longer used regularly.) The doppler radar signal, a ‘beat’ frequency
signal produced by mixing the original and reflected microwave signal with a
nonlinear mixer diode, was digitized at 1 MHz and later Fourier transformed in 512
point sections to produce a time—velocity history.

The optical velicometry system is driven by a tungsten filament lamp
through a pair of fiber optic cables. The light beams are unfocused across the barrel
extension, and only the center 0.125" of the beam is incident on the receiving fibers.
These are connected to a pair of light—activated switches. The TTL compatible
switches gate a 10 MHz oscillator crystal. A totalizing counter records the number
of cycles. The count then represents the number of 0.1 ys intervals in which the
first switch is open and the second switch is closed. An external reset prevents
counts after the first set. The distance between the beams is 4.0 cm.

The barrel vacuum is also connected to the barrel extension. This vacuum

—15—

system is independent of the recovery tank vacuum, and driven by a two—stage
mechanical pump. The barrel and powder are typically evacuated to 100 mTorr
prior to firing.

The recovery system decelerates the target containing the shocked powder as
well as the momentum trap. The momentum trap and target are independently
coupled to the rear of the recovery vacuum tank by industrial and automotive shock
absorbers, respectively. The mounts for the target and momentum trap are hinged
to the recovery tank floor. This allows relatively gentle recovery of the target. See
Fig. 2.4.

The target ring is a key component of the gun system [Fig. 2.5]. The powder
to be compacted must be contained prior to impact, prevented from flowing radially
after impact, and removed as a single piece afterward. The Keck Compactor uses a
4340 steel ring four inches in outer diameter and 2.5 inches long for this purpose.
The rings are heat treated to Rc 50. A steel liner with an aluminum plug is then
pressed into the ring. The unloading wave catches up to the initial shock wave in
the powder (before it reaches the aluminum plug). The aluminum plug is an
adequate impedance match to many metal powders, and the momentum trap
coupled to the plug is also aluminum. After the shot, the plug, compact, and flyer
plate are in the liner. The liner is pressed out and cut along the side to remove the
compact. The ring is reusable when honed out for a slightly larger liner. The ring
is typically vacuum sealed to the barrel extension to evacuate the powder, but the

powder cavity can be sealed and filled with a gas prior to loading.

2.2 Experiment

A metallic glass ribbon, Ni, ,Cr, Sig .B, -C ag, produced by Allied

76.47°19°° 2.37 1.5 ~.08°
Corporation as MBF 50, was ball milled to powder. The amorphous nature was

—16—

confirmed by x—ray diffraction. The powder was shock consolidated by a 5 mm
thick 303 SS flyer plate travelling at 1.06 km/s. These shots were initiated to
produce fully bonded metallic glasses. The powder was at 54% of full density before
consolidation, and fully densified by the shock wave. After shock consolidation, the
compact was sectioned, polished, and etched to reveal grain boundaries. The
compact is crystalline to a depth of 3.5 mm. See Fig. 2.6. This corresponds to the
position at which the shock wave pressure is reduced by the reflection from the back
of the flyer plate. Beyond this level, the powder is, in effect, consolidated by the
plastic sabot at considerably lower pressure. This is depicted in the
time—displacement diagram of Fig. 2.7. (The pressures in each zone are calculated
using the Hugoniot parameters for Ni.)

The degree of planarity in the glass—crystalline boundary is evident. Even in
the edge regions where deviations from planarity are expected, the energy deposited
by the initial shock wave remained above the threshold required to heat the
compact to the crystallization temperature. We have not carried out a detailed
calculation of the energy deposition expected in the edge regions. The appearance of
the compact suggests that the edge effect is relatively small in geometries which do
not shock the entire recovery ring (unlike solid multi—cavity fixtures such as the

Sawaoka fixtures [3]), but do contain the powder to limit radial flow.

2.3 Analysis

The position of the transition from crystalline to amorphous metal in the
compact may be determined using the Simons and Legner powder Hugoniot model
discussed in Chapter 1. The experimental parameters are summarized in Table 2.1.
Values of « and p are estimated from elemental averages. The Hugoniot is not

particularly sensitive to « and win the range of parameters used in our experiments.

—17—

Once the speed of the shock wave is calculated, the thickness of the flyer and the
sound speed of the flyer and glass are used to estimate the depth in the consolidated
sample at which the rarefaction wave originating at the flyer—sabot boundary
catches up to the shock wave. Material past this point has much less energy
deposited in it. The transition between energy deposition levels is marked by the
retention of metallic glass in the consolidated material. In the strongly shocked
region, the energy deposited is sufficient to heat the powder beyond its
crystallization temperature, 467°C. Ifthe consolidation is indeed nearly
one—dimensional, the boundary between amorphous and crystalline material will be
planar. The boundary is actually somewhat convex, due to the complex
"ring—down" at the flyer — cavity edge [Fig. 2.6]. It is far from the shallow cone
generated in an unsupported powder. A rarefaction wave originating at the flyer
plate perimeter would release the shock wave in a cone of angle 23° if the powder
were not supported by the steel ring. This angle is calculated using the approach

described in [4],

cot p= {(C/V)* — (vV-u)*/v"

where C is the relaxation wave speed in the consolidated powder (4.7 km/s), V is
the shock wave velocity (1.86 km/s), and u is the particle velocity (0.86 km/s). The
cone is not observed in the polished and etched sample. A 2—D finite element
calculation simulating the shock wave consolidation of a rigidly supported powder
by a flyer plate striking only the powder showed relatively very little edge
relaxation [5].

An energy argument supports this observation as well. In an unsupported

powder, energy dissipates radially into the unshocked powder. A rigid cylindrical

—18—

containment does not allow energy to dissipate radially very well; only elastic waves
will propagate, and they will be transmitted poorly at the powder to solid interface.
The net effect is much smaller deviation from the plane wave condition in the
powder. Note that since the steel container only interacts with the flyer plate
through the powder, the steel is shocked to below the powder — flyer plate
interaction pressure. There is no area in the compact that experiences more than

the planar shock wave pressure.

2.4 Conclusions

A propellant gun has been constructed which can accelerate a flyer plate for
shock wave consolidation and recovery of the consolidated samples.

A metallic glass has been shock consolidated to solid density and partially
recrystallized. The slightly convex recrystallization boundary indicates a nearly
one—dimensional shock condition obtained by striking only the contained powder

with the flyer plate.

Acknowledgments

Aerojet Ordnance provided the the 35mm Mann barrel in the Keck Dynamic
Compactor. The successful construction of the Compactor is due in great part to
the skills and recommendations of the Caltech Central Engineering Shop. The
Metglas experiment was carried out by Barry Krueger and Caltech undergraduate

Joseph Bach, within the research program of Prof. Brent Fultz.

—19~

Table 2.1. Parameters used to calculated shock conditions in Metglas MBF 50.

Parameter Value Units
Density 7.49 gm/cc
Sound speed 4.7 km/s

Rate of wave speed 1.45 km/s—GPa

increase with pressure
Gruneison coefficient 1.9

Isentropic compressibility .005 1/GPa

—~20—

pa
hk
fae
Lid
=>
we
oc /
rr
1] _
1 —t
/] i
alii
Lily
oat
—, << i!
Ze | FI
= (SI) lar
<= S if 1
Lad !
Ke | ja
—_ a 1
) i
yi fam)
us!) HON | 2
=> | a
— \ ”
— Th d
Co) Way
uid Way
— wt
Sy] fin
il >—
O- iri! L ]
Wt — v
x=
1 | te cS
Lid
Zz
oo oy
OQ.
Oo oO
Fig. 2.1. Cutaway scale drawing of the Keck Dynamic Compactor. The

ignition system, external vacuum systems, and barrel position drive have been
excluded for clarity. The overall length of the assembly is approximately five

meters.

—21—

BREECH BLOCK

Fig. 2.2. Photograph of the open breech with the breech sleeve attached. The
ignition wire passes through the breech block into the breech sleeve, and is
connected to a fine tantalum wire, which ignites the propellant when current heats

it.

~22—

Fig. 2.3. Photograph of the barrel extension attached to the muzzle. The fiber
optic connectors and radar antenna are attached to the barrel extension. The

vacuum connection to the barrel is also here.

—23—

Fig. 2.4. Photograph of the recovery tank interior, showing the recovery

system. The leftmost pair of hinges supports the target ring holder, connected to
the rear of the tank with automotive shock absorbers. The right hand pair of hinges

supports the momentum trap holder. The tank opening is 45 cm wide.

~24—

4540 STEEL
4130 STEEL

oe 2 50

N N
N » N
i 6061 AL
= 6 4.00" -
Fig. 2.5. Cross section drawing of the target ring. The powder to be shocked is

loaded into the ring on top of the aluminum end plug, and either pressed or sealed

into place before being loaded horizontally into the recovery tank.

—25—

{ \,

Fig. 2.6.

Photograph of the polished and etched Metglas compact. The shock
wave traversed the sample from top to bottom.

~26—

TIME VS. DISTANCE

METGLAS MBF 50
4 -
3 be
2 5
1 5
SOUD LINES ARE SHOCK WAVES
/ DASHED LINES ARE SURFACE POSITIONS
/ P=0 GPa
OF /
/ /
ISABOT , FLIER 7 | POWDER
/ , !
| he os rt i + Lt A | rm i 2 1 . i rt 3
-8 -6 -4 -2 ) 2 4 6 8
POSITION (mm)
Fig. 2.7. The one-dimensional plane wave time—displacement history

calculated for the Metglas consolidation experiment. (Edge effects are not

calculated.)

References

F. L. Williams, B. Morosin, R. A. Graham, in Metallurgical Applications of
Shock— Wave and High—Strain—Rate Phenomena, edited by L.E. Murr, K. P.
Staudhammer, M. A. Meyers (Dekker Inc, New York, 1986) p. 1013.

J. D. Jackson, in Classical Electrodynamics, (Wiley, New York, 1975) p. 521.
A. Sawaoka, Mat. Res. Soc. Symp. Proc. 24, 365 (1984).

Y. B. Zel’dovich and Y. P. Raiser, Physics of Shock Waves and High
Temperature Hydrodynamic Phenomena (Academic Press, New York, 1958)
p. 747.

T. Thomas, P. Bensussan, P. Chatagnac, and Y. Bienveu, in Metallurgical
Applications of Shock— Wave and High—Strain—Rate Phenomena, edited by M.
A. Meyers, L.E. Murr and K. P. Staudhammer (Dekker, New York, 1991).

—28—

CHAPTER 3
MELT FRACTION AND ENERGY DISTRIBUTION DURING
SHOCK CONSOLIDATION

3.1 Introduction

Many of the publications describing microstructural changes during shock
wave consolidation have pointed to microstructural changes as evidence of the
melting of material during shock consolidation [1,2,3,4]. Some are less easily
established than others; certain structures such as martensite in steel may be
created by shear deformation without melt [5]. To clearly establish the presence
and extent of melting during a shock consolidation experiment, a metallic glass
forming alloy was prepared by ball milling a rapidly solidified ribbon, annealed to
crystalline form, and then shock consolidated at a range of shock energies. When
samples of this material were polished and etched, amorphous non-etching regions
were observed. While many structural changes may accompany rapid plastic
deformation, the formation of amorphous material from crystalline material requires
either rapid solidification or (in the case of ordered alloys) very extensive plastic
deformation. (Researchers have successfully made amorphized material by both
mechanically alloying constituent materials via ball milling [6], and by mechanically
deforming an ordered intermetallic through ball milling [7]). This mechanical
process is distinct from shock consolidation in that average strains greatly exceed
unity. For a discussion of solid state amorphization transformations, see Ref. 8.
The mechanism may be a factor in the consolidation of ordered alloys. In the

Markomet 1064 (FCC structure) discussed below, it will not be considered.

—29— -

The existence and extent of melting during shock consolidation is of interc-t
here principally in the limits the extent of melting places on energy distribution
during shock wave passage. These limits are discussed in section 3.4, and address:d
again in Chapter 4. Briefly, the minimum amount of melt during consolidation
would appear if shock energy was deposited rather uniformly throughout the powder
particles. In this case temperature excursions from equilibrium are minimized.
Provided the shock energy is insufficient to raise the bulk temperature to the
melting point, no melt would be observed. On the other hand, if energy is deposited
very inhomogeneously, material can melt at shock energies far below that required
to heat the bulk to the melting point. The maximum amount of melt possible is
generated if all energy deposited goes to heating and melting a fraction of the
material, leaving the remaining material unheated.

In perfectly clean materials, good bonding via shock consolidation should be
achievable merely through full densification and cold welding. The presence of
surface contamination on powder surfaces necessitates some amount of surface
melting to effect good interparticle bonds. Knowledge of the amount of melt
produced as a function of energy applied is therefore important to predict conditions
for good bonding. The nature of surface contamination is obviously an important
variable also; the presence of hydrated oxides has been observed to interfere with
bonding in extremely energetic consolidation experiments [9].

In order to predict the amount of melt produced under several different
models of energy deposition during shock consolidation, the spherically symmetric
conduction of heat was calculated, including the motion of a solid—liquid interface.
The spatial distribution of energy most consistent with the melt observed is inferred
to best represent the actual energy distribution during shock consolidation. This

simplified model is intended to determine the degree to which energy deposition

favors particle exteriors, junctions, and asperities in preference to bulk flow.
Structural examination of the recovered material was performed using
transmission electron microscopy by Thadhani et al. [10]. The principal observations
of interest here are firstly, the confirmed amorphous nature of the non—etching
material and secondly, the presence of retained crystalline metallic borides of large
grain size in the amorphous regions. Their presence sets an upper limit on the
temperature excursion of the melt in which they were entrained; MoB, melts at

2800° C [11].

3.2 Experiments

Irregularly shaped amorphous Markomet 1064 alloy powder (Nigs gMoo, »
Crg 7Bg g, 50 um x 50 x 10 ym average platelet size) was prepared by ball—milling
melt—spun ribbons. The metallic glass was crystallized by annealing at 900°C for
two hours. See Fig. 3.1. (DTA scans revealed crystallization exotherms at 610°C
and 825°C, and melting at 1240°C.) The annealed powder was loaded into
hardened steel targets, depicted in Fig. 3.2, and pressed to 47% to 52% porosity as
listed in Table 3.1.

The targets were mounted to the end of the 20mm inside diameter (ID)
propellant driven gas gun in the Caltech Seismology Laboratory. This facility is
similar in effect to the 35mm gun described in Chapter 2. The gunpowder charge is
loaded into a shotshell. The shotshell is loaded into the breech, and ignited by a
primer cap triggered with a solenoid—driven firing pin. The charge provides
pressure to accelerate the polycarbonate sabot. The sabot is fitted with an
"O"—ring seal to prevent blow—by of propellant gas. Shock consolidation was
performed by impact with a 304 SS flyer plate supported by the lexan sabot. The

projectile velocity was varied to obtain a variety of energies. The projectile

—31—

velocities were measured by timing the interruption of two laser beams. The
hardened steel targets permitted only small radial strains. Projectile velocities
ranged from 0.96 to 1.44 km/s. Table 3.1 lists experimental parameters for the
seven shots. The target is described in more detail in Ref. 12.

The recovered samples were sectioned by electric discharge machining,
mounted, and mechanically polished. Marble’s etch was used to reveal grain
boundaries in the crystalline material. Amorphous material did not etch. A typical
region is shown in Fig. 3.3. The original particles are flattened by the shock
process, and appear to be bounded on some sides with amorphous material in rather
large zones typically 10—20 um in width, and frequently similar in size to the
particles in higher energy shots. This implies melting is not limited to a uniform
layer of particle surfaces, and may ‘jet’ into interparticle voids. In explosive
welding, ‘jetting’ of melted material is often observed [13].

The quantity of non—etching material was measured from optical
micrographs by tracing the melt pool boundaries on a digitizing tablet.
Perpendicular and parallel sections were both analyzed. In the higher energy
experiments, some of the larger non—etching regions contain very dark etching
regions, indicative of a very fine microcrystalline structure. These regions are
presumed to have melted and quenched during the shock process, but at a rate
insufficient to prevent crystallization. This material is included in the estimate of
material melted in Table 3.1. Selected area diffraction performed during
transmission electron microscopy, described elsewhere in detail [10], confirmed the
presence of amorphous and microcrystalline material in the shock consolidated

samples.

—32—

3.3 Discussion

The thermodynamic analysis of these observations begins with a compilation
of heat flow and capacity information for the Marko 1064. The parameters listed in
Table 3.2 include heat capacity, latent heat of melting, and melting point, derived
from DTA measurement. Thermal expansion coefficient and isentropic
compressibility are estimated using the values for Ni. Pressure correction for
melting temperature is similarly estimated. From this data, upper and lower
bounds for material melted can be drawn. Given the shock energies listed in Table
3.1, none of the shots reach an equilibrium temperature close to the melting
temperature of the alloy. Therefore, no melted material would be generated by
completely homogeneous heating of the target powders. This represents the lower
bound of material melted. At the other extreme, all energy could go to heating and
melting that fraction of powder which could be melted. This represents an upper
bound to the amount of material melted. More exactly, the upper bound of

material melted, Mu, can be expressed [14]:

3.1) Mu = sh
C(T,,-T,)+h

where Ey is the shock energy. Cy is the mean heat capacity, Ta is the melting
temperature of the alloy, T 0 is the initial temperature of the alloy, and L is latent
heat of melting. This upper bound, lower bound, and melt fraction data are
displayed in Fig. 3.4.

Two facts are immediately clear. First, the data conforms closely to neither
extreme of melt possible, falling squarely between them. Second, no melt at all is

observed until the shock energy exceeds a critical minimum between 214 and 333

~33—

J/g. Another important observation is the dependence of melt fraction on initial
porosity, as well as on (calculated) shock energy. The two experiments with
energies of 415 and 419 J/g have quite different amounts of melted material. Either
the shock pressure (increasing as porosity decreases at constant energy) has a strong
influence on energy deposition, or the very high porosity material is not being
shocked via the Raleigh line, and has a lower than calculated shock energy.
Insufficient data is present to distinguish between these possibilities, but they must

be considered during a more detailed analysis.

3.4 Energy Deposition Modeling

Treating the particle interactions in detail is a prohibitively complex
business at best. The irregular particles are driven into one another, mutually
shocked and deformed as the shock wave entrains them. Rapid plastic deformation
deposits thermal energy. Portions of particles melt while other parts remain
relatively cool. Porosity is removed through this disorganized process, and a dense
compact is formed. Given sufficient time under load, the compacted material cools
and bonds before release waves can knock it apart again.

All of these details are passed by in the following analysis. Only the thermal
history of the material is considered. That history is simplified by limiting the
geometry to a symmetrically heated sphere. The radius of the sphere is selected to
match the areal diameter of the particles. The resulting sphere has a thermal
response time similar to that of the original tabular particles. The behavior of the
system is now easily amenable to simulation. Assuming the thermal history of the
actual material is similar in favoring preferential heating at or near particle
exteriors, the results of the simulation will roughly correspond to that seen in the

experiments.

—34—

Thermal history simulations of this type were first used by Gourdin [14] to
examine energy localization in shock consolidation of Cu. The thermal history of a

sphere is simulated by integrating the thermal Laplacian,
Ot r Or

using a finite element algorithm based on divided differences. The fixed space

network method of Murray and Landis [15] was used to treat the moving liquid—solid

‘J

where V-is the melt front velocity, p is density, L is latent heat, In is the melt

interface. The motion of the interface is governed by

ar,
r —k-—
m Or

aT
3.3) en
pL | “dr

front position, and T 5 and T, are the solid and liquid temperatures near the
interface.

The heat is added to the sphere in several different patterns over the rise
time, and these different boundary conditions result in very different maximum melt
fractions. The four patterns are a) all heat into the particle surface, b) all heat into
the solid—melt interface; no liquid heating beyond Tm, c) all heat into the particle
bulk, biased towards the surface like the cube of radius, and d) 30% of the energy
into the solid—liquid interface and 70% into the bulk of the sphere as inc). Pattern
a) is identical to that used in [14].

To illustrate the effect of these energy flux patterns on the temperature
gradients within the sphere, the temperature vs. radius is plotted at the end of the

tise time for each case a)—d), in Fig. 3.5. Note the peak temperature in a) is

—35—

approximately 10,000 °C, while in case c) it is 900°C. In each case, the mean
energy density is 333 J/g. A physical mechanism for heating the liquid phase to
stellar temperatures is somewhat difficult to hypothesize, but this is intended as a
qualitative rather than mechanistic model.

The more interesting results of these calculations are the predicted melt
fraction vs. energy profiles, shown in Fig. 3.6 along with the measured melt data.
Both input conditions which heat the sphere via a surface flux predict the onset of
melt at 70 J/g. Heating the material through the bulk results in a high melt—onset
energy over 500 J/g, even biasing the energy towards the surface. The mixture of
surface flux and bulk heating best represents the data. The preferential heating of
asperities and inter—particle junctions is seemingly not as localized as a pure surface
flux, and the deformation energy is distributed within a significant thickness of
material. As shock energy increases, the data is well fit by profile a). From a
physical point of view, heating the melted alloy to 10,000°C is unlikely; deforming a
low-viscosity liquid is not energetically demanding relative to solid deformation.

If the shock energies in the experiments conducted at 52% porosity are
overestimated, as discussed earlier, the melt fraction predicted by pattern d) is a
better fit. Also, the melt pools implied by amorphous material in the recovered
compacts are large and poorly linked to solid interiors, relative to the melt shell
postulated in the model. The melt in the model will quench more rapidly than in
the regions observed. The amount of melt in the model is thus likely an

underestimate.

—36—

3.5 Conclusions

Shock consolidation results in highly inhomogeneous deposition of shock
energy. The formation and retention of amorphous Markomet 1064 during shock
consolidation of crystallized powder implies the degree to which shock energy is
localized. To predict both the onset and quantity of melt produced in a simple heat
flux model, energy must be added preferentially at particle surfaces, but with a
significant amount of energy expended into bulk deformation as well. In the case of
a symmetrically heated sphere, a 70% bulk to 30% surface partition of shock energy

provided the best agreement with the data.

—37—

Table 3.1. Shock conditions and quenched melt measurements of the Markomet
1064 shock consolidation experiments. Shock conditions are calculated based on the

powder Hugoniot calculations discussed in Chapter 1.

Flyer Shock Shock

Powder Vel. Pressure Energy Melt
Shot # Porosity m/s GPa J/g Measured
805 52% 962 5.1 333 0.46
807 52% 1440 * 3.4 214 0.02
808 — 52% 1090 6.3 414 0.12
813 52% 1340 9.65 592 0.28
814 52% 1188 7.58 481 0.25
815 AT% 1220 10.1 467 0.22
816 49% 1124 8.94 419 0.20

* — polycarbonate flyer plate.

Table 3.2. Thermodynamic and physical parameters for Markomet 1064 alloy.

Parameter Value Units Ref.

Density (p) 9.05 gm/cc [16]

Melting temperature (Tm) 1240 °C [17]

Average specific heat (Cv) 0.60 J/g—"C [17]

Latent heat of fusion (L) 200 J/g [18]

Average solid thermal (a) 0.055 cm? /s [19]
diffusivity

Areal diameter (D) 42 pm [16]

Pressure correction for 27 °C/GPa [20]
melt temperature (for Ni)

Isentropic compressibility 0.0057 1/GPa [21]

(for Ni)

Fig. 3.1 Photograph of polished and etched particles of Markomet 1064 powder.

BASED WASHER

G \Y; FLYER

INN: ; ;
2 LL
SSS POWDER \
ING: LEXAN

! SABOT
| TARGET RING

AXIAL SPALL
PLATE

Fig. 3.2 Illustration of the hardened steel target ring used in the shock

consolidation experiments of Markomet 1064.

—40—

Fig. 3.3 Photograph of the polished and etched Markomet 1064 compact shot
#808. Bright areas are amorphous, grey areas crystallized during the pre—shock
anneal, and very dark areas in the bright areas crystallized upon rapid quenching

during the shock consolidation.

—~41—

1 0 Lj t T Ly
0.8 F 4
Cc >
20.6 f ge 4
o &
Lo os iS
= Q
f 2
0.2 F p . ;
g >
0 S
0.0 4 i. + 1 re 2 4 I L
fe) 204 408 612 816 1020
Shock Energy (J/g)
Fig. 3.4 Plot of calculated maximum, calculated minimum, and measured melt

fraction vs. shock energy in the Markomet 1064 experiments.

(a) (b)
12.52 — 1.52 —

ms Tm

= 10.02 + 1.22

"7.52 . 92

5 5. 02 r . 62

3 .

& 2.52 t - 32

5 Tm }----------------

- 02 02

0 2 4 6 0
(c)
1,52 ——————r + 7 }

a Im t------- ee ee eH He er ee

= 1,22 + 1

= .92+

S$ .62

ee)

9 32

"02 02 _

0 .2 4 6 8 i 8 .2 4 6 -BRm
Radius? (normal ized) Radius? normalized)
Fig. 3.5 Plot of temperature vs. cubed radius at the end of the shock rise time

calculated using four different energy deposition profiles.

~43—

0.8

Melt Fraction
) 9°
> oO

ad
iN)

0 200 400 600 800

Shock Energy (J/g)

Fig. 3.6 Plot of predicted and experimental melt fraction vs. energy, using four
energy deposition profiles. The profiles correspond to: a) All heat into sphere
surface, b) All heat into solid—melt boundary (or surface if no melt), d) All heat

into the sphere volume, and d) a mixture of 30% b) and 70% c).

~44—

References

10.

11.

12.

13.
14.
15.
16.

17.

18.

19.

D. Raybould, D. G. Morris, and G. A. Cooper, J. Mat. Sci. 14, 2523 (1979).
D. G. Morris, J. Mat. Sci. 15, 116 (1981).

P. Kasiraj, T. Vreeland Jr., R. B. Schwartz, and T. J. Ahrens, Acta Metall.
32, 1235 (1984).

for example, J. E. Smugersky, in Metallurgical Applications of Shock— Wave
and High—Strain—Rate Phenomena, edited by L. E. Murr, K. P.
Staudhammer, and M. A. Meyers (Dekker, New York, 1986) p. 107.

_G. B. Olsen and M. Cohen, Metall. Trans. 7A, 1897 (1976).

for example, C. Koch, O. B. Cavin, C. G. McKamey and J. O. Scarbrough,
Appl. Phys. Lett. 43, 1017 (1983).

A. Ye. Yermakov, Ye. Ye. Yurchikov, and V. A. Barinov, Phys. Met.
Metallogr. 52, 50 (1981).

R. B. Schwarz and W. L. Johnson, ed., Solid State Amorphization
Transformations (Elsevier, Lausanne, 1988).

T. Vreeland Jr., P. Kasiraj, and T. J. Ahrens, in
Mat. Res. Soc. Symp. Proc. Vol 28 (North Holland, New York, 1984) p. 139.

N.N. Thadhani, A. H. Mutz, P. Kasiraj, and T. Vreeland Jr., in Metallurgical
Applications of Shock— Wave and High—Strain—Rate Phenomena, edited by L.
E. Murr, K. P. Staudhammer, and M. A. Meyers (Dekker, New York, 1986),
p. 247.

L. Brewer and A Lamoreaux, in Molybdenum: Physics and Chemical
Properties of its Compounds and Alloys (Atomic Energy Review, Special Issue
#7, 1981) p. 107.

T. J. Ahrens, D. Kostka, T. Vreeland Jr., R. B. Schwarz, and P. Kasiraj, in
Shock Waves in Condensed Matter, edited by J. R. Asay, R. A. Graham, and
G. K. Straub (Elsevier, New York, 1984) p. 443.

B. Crossland and J. D. Williams, Met. Reviews. 15, 17 (1970).
W. H. Gourdin, J. Appl. Phys. 55, 172 (1984).
W. D. Murray and F. Landis, Trans. of ASME. 81, 106 (1959).

The areal diameter was determined by measuring the specific area of the
powder. The major dimensions of several particles were measured from
scanning electron micrographs.

L. Lowry, private communication based on differential thermal analysis and
differential scanning calorimetry of Markomet 1064, Jet Propulsion
Laboratory, Pasadena, California, 1985.

The latent heat of melting was estimated using Richard’s rule. See D. R.
Gaskell, Introduction to Metallurgical Thermodynamics, (McGraw—Hill, New
York, 1981), p. 142. Measurements of the latent heat of melting were
attempted, but gave inconsistent results.

Estimate based on measured electrical resistivity and behavior of Nichrome:

20.

21.

—~45—

J. H. Lienhard, A Heat Transfer Handbook (Prentice—Hall, Edgewood Cliffs
New Jersey, 1981) p. 492.

D. A. Young, Phase Diagrams of the Elements (Lawrence Livermore,
Livermore, 1975).

A. B. Pippard, The Elements of Classical Thermodynamics (Cambridge, New
York, 1964).

—~46—

CHAPTER 4
SIZE DISTRIBUTION AND ENERGY DEPOSITION:
POWDER THERMOCOUPLE EXPERIMENTS

4.1 Introduction

Strain energy localization during shock wave consolidation has been
established by evidence of melted and frozen material near particle boundaries [1].
The extent of such localization can be inferred to some extent by measuring the
quantity of refrozen material [2]. This type of ‘recovery experiment’ is productive
using the materials which leave readily measurable evidence of rapid solidification
but reveals little of the dynamics of the process. A more revealing look into the
localization of shock energy requires real time measurement of temperature changes.

There are several constraints on such measurements. First, the characteristic
response time of the technique to be employed must be considerably smaller than
the shock—wave rise time in a porous medium. As established by Schwarz et al. [3],
the rise time of a shock wave, tris ein a porous medium is roughly the particle
diameter, D, divided by the shock wave velocity, V;- For a 6 GPa shock wave in
64m diameter copper at an initial porosity of 40%, the shock wave velocity is 1.6
km/s, and the rise time is about 40 ns. Second, the measurement must be made
without excessively perturbing the shock process itself, and in a region of the sample
where the shock state is simple and well—characterized. The optopyrometric
measurement of temperature at a powder—window interface, for example, adds
powder—window shock interactions to the problem. (It may nonetheless be a highly
successful technique for measuring bulk shock effects; continuum effects are well
understood, and corrections for the window impedance can be made. This technique

was used to determine the effect of pressure on the melting point of iron [4]).

~A7—

These criteria may be met utilizing the thermoelectric (or Seebeck) effect.
Electric potential is generated by a thermal gradient in a conductor. For my
purposes, the conductors are metals with electrons as the charge carriers. Roughly
speaking, higher temperature metal has higher velocity electrons. They tend to
diffuse more readily than the lower velocity electrons in the colder region. This
creates a net excess charge in the colder region, and a positive voltage from hot to
cold ends of the conductor. More accurately, a higher temperature results in a
larger number of electrons occupying energy states above the Fermi level in the
metal. These slightly more energetic electrons have a higher mean velocity and
their higher mobility creates a net electron current (or net voltage gradient, if the
circuit is open). The effect is monotonic and roughly linear over a wide temperature
range in many metals. Dissimilar metals generally have different thermoelectric
behavior. If two metals with different thermoelectric coefficients are joined, the
temperature at the junction may be determined by measuring the electric potential
(or EMF) across the ends of the wires, provided the temperature of the ends (the
’reference temperature’) is known. A more complete discussion of the
thermoelectric effect may be found in Solid State Physics, by Ashcroft and Merman
[5]. These thermocouples’, fabricated from standardized alloy pairs such as
platinum/platinum with 10% rhodium, and copper/copper with 45% nickel
(constantan) are commonly employed to measure temperature.

Embedding a conventional thermocouple in a powder would create
impedance mismatches and other problems limiting the validity of the data during
the rise time. If, however, the thermocouple is formed by the powder itself, this
problem is avoided. More specifically, two powders of very similar shock impedance
but different thermoelectric strength are layered parallel to the shock front. The

EMF generated by this powder—powder thermocouple is recorded. Copper and

—48—

constantan were used in these experiments. The effect of varying particle size
distribution on thermoelectric output was explored using this technique.
Experiments measuring shock—generated EMF were first made in 1959 by
Jacquesson [6]. Thermocouples were imbedded into the material of interest, and
EMF recorded. The difficulty lies in the extreme locality of the particle—particle
interactions. The shock impedance and thermal response of the powder differed
considerably from that of the thermocouple In addition, a host of anomalous
voltages can be generated via friction between thermocouple elements, triboelectric
effects, and electrical impedance problems. In spite of these difficulties, Bloomquist
et al. successfully used a diffusion bonded thermocouple pair imbedded in a shocked
media. This resulted in accurate equilibrium temperature measurements as
predicted by Hugoniot calculations [7]. Nesterenko first demonstrated that the
shocked powders themselves could form the thermocouple [8]. Schwarz et al. used
copper and constantan powders to form the thermocouples, and measured the
rise—time of the shock wave to be similar to the time required for a shock wave to

traverse a single particle {3}.

4.2 Thermocouple Experiment Design

Since the object of this work was to explore the effect of size on energy
distribution, powder size and shape uniformity was of primary importance.
Characterizing the behavior of an irregular powder would be more difficult, since
several spatial measurements of the powder would be necessary, so a nearly
spherical powder was sought. Truly mono sized spherical powder would be ideal,
but powder fabrication processes rarely achieve this ideal. (Polymeric sphere
standards are produced, and the carbonyl] process can produce highly spherical metal

powders.)

—49—

Given the required thermocouple alloy compositions, we elected to procure
inert—gas atomized copper and constantan powders from Ames Laboratory in Ames,
Iowa, and HJE Corporation in Troy, NY, respectively. The copper powder was
atomized from pure copper (CDA101), and delivered sieved into size fractions. The
constantan was atomized from thermocouple stock ingot from Omega Engineering,
Inc. of Stamford, CT. The constantan exhibited poor electrical conductivity, as
received, and was treated in wet hydrogen at 400°C for 2 hrs to reduce surface
oxides. The constantan powders were sieved using a mechanical sieve shaker and
bronze mesh screens into size fractions from 37-44 yum, 44—53yum, etc. Photographs
of the powders are presented in Fig. 4.1.

The next step was designing a thermocouple target fixture and loading
protocol which would insulate and shield the thermocouple circuit, and allow the
thermocouple to be built. By fabricating the target out of copper, thermoelectric
potential difference between the target and the copper powder were avoided. This
was confirmed in the shock experiments; no EMF was measured as the shock wave
traversed the copper plate—copper powder interface. The target consists of a
circular plate counterbored to fit onto the barrel extension at the end of the gun
muzzle (see Chapter 2), with a short copper tube soldered onto the flat side of the
disc. A quartz sleeve is slipped into the copper cylinder, providing electrical
isolation within the cylinder. See Fig. 4.2.

First, the copper powder was loaded to a depth of one mm, and leveled with
a depth micrometer. Next, an annular disk of ceramic felt was placed onto the
copper. The ceramic (Cotronics Corp. Brooklyn, NY) limits the thermocouple
contact to the center 5 mm of the disk. The shock wave is planar in this region of
the powder; edge interactions originating at the flyer plate edge remain outside the

center. The periphery of the disk is covered with 44m alumina powder to insure

—50—

insulation at the edge. Three mm of constantan powder is next loaded into the
target. The constantan is also leveled. Several mm of irregular copper powder is
placed on the constantan to provide an easily compressible contact layer. The
porous bronze cap is pressed onto the copper, and epoxied to the quartz sleeve under
several hundred pounds of pressure. (The load is increased until the thermocouple
resistance is less than one 9.) See Fig. 4.3. EMF is measured between the bronze
plug and the copper cylinder.

The copper sleeve provides enough support to allow light compression of the
powder without fracturing the quartz. Since retention of the initial powder shape
was desired, I wanted to avoid pressing the green compact strongly. This meant
some amount of static compression had to be maintained on the compact to retain a
low resistance through the powder junction. The epoxy bond between the bronze
plug and the quartz, cured under load, provided this compression.

The electrical circuit had to combine high input impedance and low
inductance, to deliver true thermoelectric potential values with nanosecond response
time. Initially the thermocouple circuit was simply connected via an RG58 BNC
cable to the 500 termination of a Tektronix 11302 oscilloscope. The inductance of
the BNC cable, combined with the low impedance, pulled the signal down to zero
after a few unrevealing oscillations. A high speed op—amp video buffer—follower
(National Semiconductor LH0063CK) was used to construct a robust voltage
follower capable of supplying the oscilloscope with an accurate signal. A short
length of micro—coax cable connected the thermocouple to the voltage follower.
This circuit was tested with a 500 MHz triangle wave for frequency response and
gain accuracy. The signal was recorded and digitized using a CCD camera attached
to the microchannel plate display screen of the oscilloscope.

To trigger the oscilloscope at the proper time, an enameled 0.1 mm dia.

—51—

copper wire was glued across the impact face of the copper target plate. The wire
was kept at a 750 mV potential, and was broken or shorted to ground by the impact
of the flyer plate. The output of the wire was connected to the oscilloscope trigger
with a 500 termination.

The thermocouple targets were placed on the barrel extension, supported by
an aluminum ring against the momentum trap. The aluminum ring also served as a
debris containment, and a connector mount for the trigger wire. A 25mm dia. 303
SS flyer plate, mounted on a 35mm nylon sabot, was accelerated down the
evacuated barrel into the thermocouple target plate. The shock wave generated by
the impact was transmitted through the copper plate and into the powder

thermocouple. The targets were, of course, each destroyed by the experiment.

4.3 Experimental Results

The intent of these experiments was to elucidate the degree of energy
segregation in a powder mixture as powders of different size are shock—consolidated.
If a sizable portion of the powder melts upon impact, the details of the rise time and
initial temperature decay are lost in the temperature plateau created at the melting
point. The velocities employed were chosen at a level generally inadequate for full
interparticle bonding (though sufficient for full densification). The first experiment,
with a flyer velocity of 872 m/s, resulted in a 100 ns voltage plateau at 75+2 mV.
See Fig. 4.4. The uncertainty is a result of a non-steady zero—level and 8—bit data
range. This should correspond to the EMF generated by the copper—constantan
couple at the melting point of copper under a pressure of 5.3 GPa, assuming
material is not heated beyond its melting point during shock consolidation. (This
assumption was discussed in Chapter 3.) At 5.3 GPa, copper melting point is

elevated to 1260°C based on interpolation of the results of Akella [9].

—52Q—

Thermocouple calibration was based on data published by Bartels et al. [10], and
linearized to 6.5 wV/°C; the data is not quite linear, but the interest here is in the
dynamic response rather than the exact temperature value. See Fig. 4.5. 75 mV
corresponds to a temperature of 1175°C, without correcting thermoelectric EMF for
pressure. Since the thermoelectric signal is an average of the EMF at all points on
the copper—constantan interface, a voltage plateau below that corresponding to the
melt temperature can still indicate the presence of some liquid.

The thermocouple shots are summarized in Table 4.1. A range of powder
size combinations and energies was explored. The digitized traces are displayed in
Fig.’s 4 and 6-10. An example of the raw data (captured from the CCD camera) is

shown in Fig. 4.11.

4.4 Discussion and Analysis

The analysis of this data (for elucidating the energy deposition in the
powders as a function of particle size) requires analysis of the signal during the rise
time and subsequent decay during the characteristic temperature relaxation time of
the particles involved. Several effects interact to create the output shape. Copper
and constantan spheres are in contact, thermally and electrically conducting, and
mutually deforming as the shock wave consolidates and heats the powder particles.
The thermocouple interface is composed of 25 mm” of powder particles in contact.
The interface is flat in only the roughest sense; it is approximately parallel to the
shock plane. Irregularities characteristic of at least the radius of the larger particles
are expected. In addition, a certain amount of mixing is possible, especially when
large and small particles are combined. Some amount of average tilt is expected
too, though efforts were made to control tilt to within a few degrees.

A realistic simulation of the above phenomena would require a

—53—

three—dimensional finite element analysis of daunting size. More limiting is the
absence of a tested model for material deformation at strain rates over 10" /s. This
work is limited to an examination of the thermal deposition and conduction which

occurs during the shock wave passage. The heat conduction equation,
4.1) at = 49?

was applied during and after the passage of an energy input front representing the
shock wave. Solid density was assumed throughout. Even this problem becomes
daunting when the lack of a symmetry direction, change in thermal properties, and
melting of particle surfaces are included. Further simplifications render the problem
tractable, although the conclusions are necessarily somewhat qualitative as a result.
The simplest dynamic representation of a particle heated preferentially on
the outside consists of two thermal masses, the particle core and crust, connected by
a thermal resistance giving the system the relaxation time characteristic of a sphere.
The core, or inner mass, is an inner sphere concentric with the particle. The crust,
or shell, is the remaining hollow sphere formed by subtracting the core from the
original particle. Next, the ratio of the shell mass to the core mass, defines the
relative sizes of these thermal elements. For the moment, allow the sphere no
melting point. If many such spheres are in contact (allow them to flatten to contact
one another), then the crusts will conduct to one another while the cores each
conduct to only their respective crust. Note there is no net heat conduction parallel
to the shock plane; simplify the picture by reducing the model to a line of particles
like a string of pearls along the shock direction. Connect them thermally as before.
Now ’blur’ them (convolve them spatially by at least one particle diameter) to allow

for random stacking of the many particles near the interface and the roughness of

—54—

junction itself. The ’blurring’ removes the periodic oscillation of temperature
created by the discrete particles, but still allows a net heat flow across the interface.
The ’blurred’ string can be simulated with relative ease.

The junction was modeled as a one-dimensional string of thermal masses
conducting heat to one another and each to a single internal thermal mass. See Fig.

4.12a. The continuum differential equations for this system (not at the junction)

are:

dTo_ do, ym
4,2) in = aw + 1(Ti—To)
4.3) GU = m(To-Ti)

with To the particle shell temperature, Ti the particle interior temperature, and k,
1, and m the thermal diffusion coefficients linking them. This system of equations is
not analytically simple when the coefficients change abruptly and the particle shells
start to melt.

The thermal properties change at the junction between copper and
constantan. This may be thought of as representing a continuous copper wire
butted to a constantan wire. The wire has a core which conducts only radially. The
shock wave is reduced to a heat wave traveling at the shock velocity down the wire,
depositing energy mainly into the outer wire, as a shock wave deposits energy
mainly into the outside of a particle See Fig. 4.12b. In addition, the heat wave can
be perturbed to deposit energy preferentially into either of the materials near the
junction. Average energy deposition is unchanged. |

The final step is to allow for partial melting of the particle crust. The crust

here has a mass of half the entire particle. A simple energy sink was chosen to

—55—

represent the melting transition; at each node a record of melted material is kept. If
a node temperature would exceed the melting temperature, the energy is placed into
this record. Upon cooling subsequent to shock wave passage, the energy is
conducted out of the melted material before any solid is cooled. The simulation
stops if any particle crust is entirely melted.

As the simulation runs, the junction temperature is recorded. Following the
end of the run, the resulting temperature history of the junction is convolved with a
spreading function to simulate the tilt and roughness of the powder junction. The
tilt of the junction is represented by a square wave convolution. Mixing is
represented by a triangle wave convolution. This system was simulated and run on
a PC—compatible computer. The model code, written in Microsoft QuickBasic, is
listed in Appendix B.

Averaged thermodynamic values of thermal conductivity, diffusivity, and
density are listed in Table 4.2. The melting point of copper was corrected for
pressure. The melting point of constantan was fixed to match that of copper. The
parameters which can be varied in the model are:

A) the percentages of shock energy into the crust and core of the elements,

B) the relative masses of the crust and core,

C) the degree to which energy is preferentially deposited into the copper and

constantan near the powder junction, and

D) the width of the convolution of the thermocouple output with a triangle

function representing junction roughness.

These parameters allow sufficient freedom to fit almost any likely thermoelectric
output. Varying all of them to achieve a best fit in each case is not the intent of
this study. Rather, a reasonable determination of each parameter from the

collective behavior of the experiments was attempted as follows.

—~56—

Given the similarity of the experiments carried out, the gross character of
the energy deposition was not expected to change. Thus parameters A) and B) were
held fixed for all of the simulations. Using the duration of the voltage plateau in
shot #77 (Fig. 4.4), and the shock energy required to melt material to form such a
plateau as guides, parameters A) and B) were set at 70:30 and 1:2 respectively. If
more energy is input into the particle core, material will not melt to form the
voltage plateau. If the particle crust is more massive, melting is similarly
suppressed.

In each shot, a somewhat different degree of thermocouple planarity existed.
The mixing of copper and constantan particles was enhanced when the size
difference was very large; small particles fell into the interstices between large
particles. Tilting of the thermocouple junction relative to the shock plane varied
_ Slightly in each experiment. These effects directly influence the width and shape of
the initial voltage rise of the thermocouple signal. In each experiment, parameter
D) was determined by fitting this portion of the thermocouple signal. Note that the
effect of thermal relaxation during the rise time (approximately 100 ns) is minimal;
given a thermal diffusivity of 1 cm?/s, the characteristic thermal distance is 3 ym.

Finally, parameter C), the shock energy distribution near the thermocouple
junction, was varied as shown in Fig. 4.13. The shape of the initial voltage decay
from maximum is sensitive to this variable. The energy is biased into the copper or
constantan with the saw tooth profile since this conserves average energy as
demanded by the Hugoniot. This is the major parameter of interest in this study of
the effect of particle size distribution on energy deposition. Parameter C) was
varied to provide the best fit, especially to the initial decay slope from the voltage
maximum; the accuracy of the absolute value of the maximum voltage is expected

to suffer somewhat from the lack of temperature—varying thermodynamic

—57—

parameters. The model output is displayed with the data in Fig. 4.4, and 4.64.10.
In Fig. 4.10, note the result of varying parameter C) on the output.

As illustrated by this model, particle size strongly influences energy
distribution. The fine powder, copper or constantan, is consistently heated in favor
of the larger powder. In the first thermocouple experiment, shot #77, the wider
distribution of constantan powder size (and smaller minimum size) results in more
energy deposited into the constantan. The effect is not as great as in the more
extreme size disparities of shots #108 and #109. Secondarily to particle size,
hardness influences energy deposition. The energy bias into the copper for best fit
in shot #91 (Mext = 1.0) is larger than the bias into the constantan in shot #109
(Mext = —0.5), though the ratio of sizes is similar; the copper to constantan ratio in
shot #91 was 5:9, and in shot #109 was 10:4.

The simulation provides a poor prediction of system behavior when the
particle sizes differ by much more than a factor of 2. The behavior of shot #108,
with a size ratio of 1:3, was not well predicted. This may be attributed to a high
degree of particle intermixing at the interface, which would reduce porosity and
total energy deposited near the interface.

The thermocouple simulations of shots #80 and #109 provide somewhat
contradictory results; though particle sizes are similar, energy in shot #80 appears
to be biased towards the larger copper particles (Mext = 0.25), while in shot #91
energy appear to be preferentially deposited into the constantan (Mext = —0.5). A
certain degree of melting in shot #80 (E = 210 J/g) may be masking the detail in
shot #80, causing an anomalously slow decay from maximum slope. A voltage
plateau in shot #108 (E = 215 J/g) was observed. It would be interesting to repeat

several of these experiments at somewhat lower shock energies.

4.5 Conclusions

These experiments have demonstrated (albeit qualitatively) the preferential
deposition of energy in small particles in contact with larger powder particles during
the shock wave consolidation process. In addition, and to a lesser degree in the
hardness range tested, softer particles absorb energy preferentially to harder
particles.

The energy deposition process during shock wave consolidation of spherical
particles has been modeled with a one—dimensional simulation including preferential
energy deposition at particle surfaces, preferential energy deposition into smaller
particles, conduction into particle interiors, conduction between particles, and

incomplete melting and freezing of particle shells.

—59—

Table 4.1. Powder thermocouple experiments summary.

Cu Const. Flyer Shock Equil. Maximum Rise

Size Size Vel. Press. Temp. EMF Time
Shot (um) (um) (m/s) (GPa) (°C) (mV) (ns)
77 74-88 53-106 872 5.3 680 75 40
80 90-106 53-63 788 5.0 550 59 90
91 44-53 74-106 826 5.5 590 53 110
92 63-74 63-74 818 5.4 580 68 37
106 18~38 74-106 = 260 0.7 90 5 130
108 18—38 74-106 799 5.2 545 60 90
109 90-106 37-44 726 4.4 470 60 95

Table 4.2. Copper and constantan thermodynamic data used in model.

Property Copper Constantan
: 3

Density g/cm 8.9 8.9

Thermal Conductivity W/m-—s—K 385 35

Thermal Diffusivity em2/s 1.11 0.101

Thermoelectric Power pV/[°C 6.5

Melt Temp. at 1 Bar °C 1083 1083

dTm/dP C/kBar 3.36 3.36

Latent Heat of Fusion J/g 205 205

—60—

Table 4.3. Powder thermocouple modeling parameters. Powder sizes, shot energy,
and melting point are measured or calculated. The concentration of shock energy in
outer vs. inner portions of particles, and relative mass of shell to core of particles are
fixed for all cases. Energy bias near junction to copper, tilt convolution width, and
mix convolution width are fit to each case.

Cu Con. Melt Eext Mext Ehbias Tilt Mix
Dia. Dia. Etot Temp over over toCu Width Width
Shot (wm) (um) J/g °C Eint Mint @junc (ym) (yum)
77 80 80 260 1280 860.7 0.5 —0.25 0 40
80 98 58 210 1250 =0.7 0.5 +0.25 37 25
91 50 90 230 1250 =0.7 0.5 +1.0 50 12
106 =. 28 90 29 1100 =0.7 0.5 +05 14 14
108 28 90 215 1250 =0.7 0.5 +0.25 0 42
109 = 98 40 185 1250 0.7 0.5 0.5 25 25

Constantan.

Fig. 4.1 Photographs of the copper and constantan powders, sieved to

63-74 ym.

~62—

pt )()_] mm) ———

Porous brass plug

Copper powder

Copper = Constantan powder M9 mm
Quartz
Alumina powder ~
Ceramic felt _1 [> P l [ 4
Copper powder a7

7 “ i 7 VA " / 7 . /
SL2 Witt: ZLLLLMLL LL. 04 SLL LLL

Copper plate

Fig. 4.2 Illustration of the thermocouple target, showing the powder layers.

—63—

HEH Fitrpneaspsosageay yas
pp
wag tre fa iz

Fig. 4.3 Photograph of the thermocouple target, assembled with a hole in the

bronze plug for the signal output wire.

~64—

90 a a ee a ee ee ee ee
> .
E 79T
Ss L
a L.
Se 50}
o .
6 L
3 L
9 30+
E .
5 . 4
aay r 4
- 1 O ~~ I ~
5 I 4
_ -—-—/ 4
—1 O ae a a a a a cc Se ee a SO eS aN a
O 400 800 1200 1600 2000
Time (ns)
Fig. 4.4 Plot of thermocouple voltage vs. time, from shot #77, and the result

of the thermocouple simulation program with parameters listed in Table 4.3. The
voltage drop at 1700 ns corresponds to the shock wave arrival at the second copper

powder layer; the constantan layer was 1.75 mm thick. (See Fig. 4.2).

~65—

80 ‘ T r t * T r T’

60 Ff
— 40 }
is

20 +

QO 2 1 rn i a lL 2 1 n
0 200 400 600 800 1000
Temperature (C)
Fig. 4.5 Plot of EMF vs. temperature for a copper—constantan thermocouple

with a 20°C reference temperature, from data published by Bartels et al. [10].

~66—

90 Rr eet

7OF

Thermocouple Output (mV)

Fig. 4.6 Plot of thermocouple voltage vs. time, from shot #80, and the result

of the thermocouple simulation program with parameters listed in Table 4.3.

~67—

90 t Li T i i EJ T J LI T t i Li t LI i LT LJ T

70 F

Thermocouple Output (mV)

Fig. 4.7 Plot of thermocouple voltage vs. time, from shot #91, and the result

of the thermocouple simulation program with parameters listed in Table 4.3.

~68—

1 4 a SO a eee ee es a ee ees ee ee ee ee ee ee ee
€ 11+ ;
st
= ft
a 8F _-
x) r :
QO
3 .
8 9
E L
< a

—1 L 1 n | 1 i 1 | 1 i a a 1 1 r 1 i i L
0 400 800 1200 1600 2000
Time (ns)

Fig. 4.8 Plot of thermocouple voltage vs. time, from shot #106, and the result

of the thermocouple simulation program with parameters listed in Table 4.3.

-~69—

90 i s,s a Rees Se a eee ees ee ee ee es ee ee
= _ |
—€ OT
Ss .
io L
a 50}

5 90]

vot 5

=) .

2 30}

< -
t -

800 1200 1600 2000

Time (ns)
Fig. 4.9 Plot of thermocouple voltage vs. time, from shot #108, and the result

of the thermocouple simulation program with parameters listed in Table 4.3.

~70—

90 e——— EE
——— Data |
~--- No Energy Bias
> es Bias into Con 1
ce /OF -—+— Bias into Cu 4
2. .
2 50-
v .
a L
am) .
Q 30 r
E :
. L ,
- 10
a O 1 1 1 1 rn i. 4 | 1 m 1 l i r" n i 1 allen i
O 400 800 1200 1600 2000

Time (ns)

Fig. 4.10 Plot of thermocouple voltage vs. time, from shot #109, and the result
of the thermocouple simulation program with parameters listed in Table 4.3 (labeled
Bias into Con). In addition, the results of the simulation assuming no energy bias

and energy bias into copper are shown.

_71~

es
RRS

Pied,

CCD camera image of thermocouple voltage vs. time from shot #109,

Fig. 4.11

with thermocouple voltage direction negative.

—~72—

a) Discrete Representation
of Thermocouple Model

Shock Direction OS

Cu Cu Cu | Con Con Con Con Con
Shell Shell Shell t—phell hell ;}—hell |}-—Shell |}—fhell
C C C |
Core Core Core | Con Con Con Con Con
| Core Core Core Core Core
THERMOCOUPLE
JUNCTION
Core y Note: core only conducts radially.
Copper Constantan

\— Shell

b) Continuous Representation
of Thermocouple Model

Fig. 4.12 Schematic representation of the thermocouple model. In a), the
discrete representation, thermal masses are represented as boxes, and thermal
resistances as lines. In b), the continuous representation, the core does not conduct

in the axial direction.

500 , 7 , ? r , ,
Copper -: Constantan

400 F
oO
oO
™~
x 300 AN
re
[e)
al

200 F

Bias Into Copper
— — 7 Nobias
‘== + Bias into Const.
1 00 1. : . L . : 1 rT 1.
0 20 40 60 80 100
Depth (um)

Fig. 4.13 Plot of energy flux vs. position as applied in the thermocouple model,
under positive, zero, and negative energy bias into the copper. These correpond to

Eext = 1, Eext = 0, and Eext = —1, respectively.

~74—

References

10.

N. N. Thadhani, A. H. Mutz, P. Kasiraj, and T. Vreeland Jr., in Metallurgical
Applications of Shock— Wave and High—Strain—Rate Phenomena, edited by L.
E. Murr, K. P. Staudhammer, and M. A. Meyers (Dekker, New York, 1986),
p. 247.

T. Vreeland Jr., P. Kasiraj, A. H. Mutz, and N. N. Thadhani, in Metallurgical
Applications of Shock- Wave and High—Strain—Rate Phenomena, edited by L.
E. Murr, K. P. Staudhammer, and M. A. Meyers (Dekker, New York, 1986),
p. 231.

R. B. Schwarz, P. Kasiraj, and T. Vreeland Jr., in Metallurgical Applications
of Shock—Wave and High—Strain—Rate Phenomena, edited by L. E. Murr, K.
P. Staudhammer, and M. A. Meyers (Dekker, New York, 1986), p. 313.

Williams Q., Jeanloz R., Bass J., Svendsen B., Ahrens T. J., Science. 236,
4798 (1987).

N. W. Ashcroft and N. D. Merman, Solid State Physics (Saunders,
Philadelphia, 1976) p. 256.

J. Jaquesson, Bull. du G.A.M.A.C. IV. 4, 33 (1959).
(a 7) ast G. E. Duvall, and J. J. Dick, J. Appl. Physics. 50, 4838
1979).

V. F. Nesterenko, Paper presented at the All-Union School on Explosion
Physics in Krasnoyarsk, USSR (1984). English translation by the Berkeley
Scientific Transl. Service, #UCRL—-TRANS—12054, March 1985.

Akella J., J. Geophys. Chemistry. 76, 4969 (1971).

J. Bartels, P. T. Bruggencate, H. Hausen, K. H. Hellwege, K. L. Schafer, and
E. Schmidt, ed., Landolt—Bornstein: Zahlenwerte und Funktionen aus Physik,
Chemie, Astronomie, Geophysik, und Technik (Springer—Verlad, Berlin, 1957)
p. 47.

—75—

CHAPTER 5
HARDNESS AND DEFORMATION IN SHOCK CONSOLIDATION

5.1 Introduction

Material strength is one of the most obvious fundamental parameters
influencing the shock wave consolidation process. The effect of strength on
heterogeneous energy deposition is not well established. Another complication is the
lack of strain rate hardening data at very high strain rates, as discussed in the
introduction. Notwithstanding these limitations, an attempt to investigate the
effect of strength on energy localization has been made; will a harder material
experience more uniform or less uniform heating during consolidation? The
hardness of the material has been used to characterize the strength.

Observations of the effect of hardness on shock wave consolidation can be
complicated by differences in particle shock impedance, size, and melting behavior.
To isolate the effect of hardness, a highly hardenable steel alloy, M350 maraging
steel, was used as a study material. By heat treating for different times at 480°C,
the yield strength may be varied from 0.8 GPa to 2.4 GPa [1]. By shock
consolidating multiple samples of the powder in the same shot, very similar impact
conditions were achieved. The M350 powders were shocked in the hardened and
unhardened states, and with hard and soft powders mixed together. The samples
were examined metallurgically, tested for post—-shock hardness, and tested
mechanically in tension using the miniature ‘dog bone’ tensile test apparatus

described in Chapter 6.

—76—

5.2 Experimental Design

M350 maraging steel (Ni, gMo, Cojo pTiy gAly 1) powder, prepared via
inert gas atomization, was obtained from United Technologies. The powder
particles are roughly spherical, as shown in Fig. 5.1. The powder was sieved into
various size fractions, and some of the powder was heat treated in Ar for 6 hours at
480°C. Under this heat treatment, the hardness of M350 steel typically increases
from HV 280 to HV 700 [1]. In the first set of experiments, the hardened and
unhardened powders were loaded into cavities formed by drilling 10mm holes into
porous bronze filter material. See Fig. 5.2. The bronze (from Pacific Sintered
Metals) has a solid density of approximately 8.4 gm/cm®, and the maraging steel
has a density of 8 gm/ cm. The impedance mismatch between the two materials is
sufficiently small to retain planar shock conditions over the majority of the
consolidated slugs. (A truncated cone of near right angle will be consolidated under
the plane wave condition, as discussed in Chapter 2.) Experimental parameters are
summarized in Table 5.1. In the second set of experiments, the powders were placed
into three 120° sectors of the 32 mm dia. target cavity. The sectors were defined by
0.1 mm thick 304 SS sheets. Since the sheet thickness was comparable to the
powder diameter, strong perturbations in the shock state were avoided far from the
sheet; the powder ‘rings down’ to the average shock state. To reduce surface oxides,
the powders were treated in Ho at 350°C for 2 hrs.

Vickers hardness measurements were made using a Leitz Miniload hardness
instrument, loading the diamond indenter with 50 gm. The indents were typically

15 pm on the diagonal.

~77—

5.3 Results and Discussion

After shock consolidation, the compacts were mounted, polished, and etched
with Marbles reagent to reveal grain boundaries. The small compacts produced in
the first experiments showed non—etching regions at particle peripheries which
appear to correspond to melted and rapidly quenched material. See Fig. 5.3. The
larger compacts were shocked at higher energy, and did not exhibit this structure
except near the compact edges where quenching by the target ring was most rapid.
See Fig. 5.4. A mixture of hardened and unhardened powders was shocked also.
The polished and etched structure in shown in Fig. 5.5. The hardened particles etch
darker, due to presence of intermetallic precipitates on the boundaries of martensitic
laths [1]. Note the greater deformation of soft particles at the intersections of hard
and soft particles. When particles of similar hardness interact, they do so as in Fig.
5.3; the local neighborhood of particles determines the deformation pattern.

Vickers hardness, HV, is defined as

where d is the length of the indent diagonal in microns, and M is the indent load in
grams. Within a system, the hardness is roughly proportional to strength, but
different work hardening behaviors will substantially change this. The ratio of
strength to hardness is not exact, and may change from material to material. A
more complete discussion of various hardness tests is included in Smithells Metals
Reference Book [2].

The hardness of the particles was altered during shock wave consolidation.
In particular, the softer particles were hardened during the shock consolidation

process. Since maraging steels have very little work—hardening behavior [1], the

—~78—

temperature excursion during consolidation is most likely to be responsible. The
non—etching regions had a hardness of HV 420 — 570. The thickness of the melt
pools is difficult to measure, and the impressions may be of depth comparable to
pool thickness in the lower hardness measurement. The hardened particles had HV
400 +40, and the unhardened particles had a post—shock hardness of HV 320 + 20.
The particles were identified by etching behavior, and etching the grain boundaries
of the hardened particles apparently reduced the measured hardness. Wrought
M350 maraging steel in the unhardened condition has a typical hardness of HV 280,
and upon heat treatment is brought to HV 700. The broken tensile samples were
also hardness tested. The tensile samples had hardness of HV 550 + 40.

Tensile tests conducted on dog bone shaped samples of the shock
consolidated and heat treated material were conducted. The sample were machined
using an Agietron electric discharge machine. See Chapter 6 for details of the
tensile apparatus. A variety of heat treatments were made. See Table 5.2. The
samples all exhibited incomplete particle bonding, with ductile failure occurring at
well bonded particle junctions, as exhibited in the SEM fractograph, Fig. 5.6. This
is in contrast to the complete bonding of Pyromet 718 discussed in Chapter 6.
Following these tensile tests, higher energy shots were made to improve interparticle
bonding. Neither a reducing treatment in hydrogen nor larger shock energies
changed this behavior.

The lack of complete interparticle bonding is puzzling. A variety of other
steels, such as 304SS [3] and AISI 9310 [4] have been well—bonded using shock wave
consolidation under comparable conditions. The amount of melted and refrozen
material, as characterized by the fraction of nonetching material, is similar to that
in the successful consolidation of Markomet 1064, as discussed in Chapter 3. The

powder surfaces may have been contaminated or oxidized, but the powders as

—79—

received are bright, and maraging steels are corrosion resistant relative to low—alloy
steels. Surface contamination, a traditionally bothersome problem during shock
wave consolidation must be considered the most likely culprit, but it is possible that
a very high flow strength at high strain rate would tend to delocalize energy
deposition during shock wave consolidation and interfere with good interparticle

bonds.

5.4 Conclusions

Within the maraging steel system, uniform particle hardness does not
measurably change the energy deposition process during shock wave consolidation.
Shock pressures well in excess of the material yield strength were employed. The
quality of the resulting compacts did not change under a threefold change in initial
yield stress.

When hard and soft particles interact under shock conditions, soft particles
are significantly deformed relative to neighboring harder particles. Only directly
interacting particles are obviously affected; this is consistent with a particle scale
width shock front as discussed in Chapter 4. Energy input into soft particle

interiors is evidently higher.

Table 5.1. Parameters in M350 maraging steel shock consolidation experiments.
Shock pressure and energy are calculated using Hugoniot data from LASL Shock
Hugoniot Data [5], for Vascomax 300 maraging steel.

Flyer Shock Shock Powder Powder
Velocity Pressure Energy Size Heat
Shot Porosity (m/s) (GPa) (J/g) (um) Treat
16-1 31% 990 9.3 260 74-150 as—rec.
16-2 31% 990 9.3 260 75—150 6hr 480°C
16-3 31% 990 9.3 260 75—150 50% as—trec.
50% 6hr 480
17-1 33% 1240 12.7 395 75—150 as—rec.
17-3 33% 1240 12.7 395 75—150 6 hr 480°C

Table 5.2. Results of tensile tests on small ‘dog bone’ tensile samples of M350 steel,
prepared by shock consolidation and by hot isostatic pressing, after various heat

treatments.
Powder © Post—Shock Failure Stress
Heat Heat not ductile)
Shot Treat Treat GPa)
17-1 none 5 hrs 490°C. 1.11
none 1 hr 815°C and 1.25
5 hrs 490°C
none 5 hrs 490°C 1.43
17-3 6 hrs 480°C 5 hrs 490°C 1.21
6 hrs 480°C 5 hrs 490°C 1.21
6 hrs 480°C 1 hr 815°C and 1.25

5 hrs 490°C

Fig. 5.1 Scanning electron micrograph of M350 maraging steel powder, 63—74um

powder size.

Fig. 5.2 Photograph of porous bronze target insert. Holes are reamed to 10mm

dia., thickness is 9.5 mm, and outer diameter is 32.2 mm.

—83~

Fig. 5.3 Micrograph of polished and etched shock consolidated M350 maraging

steel powder (shot #17—3), heat treated before shock.

—84—

Fig. 5.4 Micrograph of polished and etched shock consolidated M350 maraging

steel powder, from near edge of large compact (shot #62).

—85—

Fig. 5.5 Micrograph of polished and etched shock consolidated mixture of

hardened and unhardened M350 maraging steel powders (shot #16—3)..

—~86—

Fig. 5.6 Scanning electron image of fracture surface of tensile sample #1 in

Table 5.2, heat treated before and after shock consolidation.

—87—

References

1. B.P. Bardels, ed., Metals Handbook Ninth Edition Vol. 1 (ASM, Metals Park
Ohio, 1978) p. 449.

2. E. A. Brandes, ed., Smithells Metals Reference Book, Sitth Edition
(Butterworths, London, 1983) p. 21-1.

3. G.E. Korth, J. E. Flinn, and R. C. Green, in Metallurgical
Applications of Shock— Wave and High—Strain—Rate Phenomena, edited by L.
E. Murr, K. P. Staudhammer, and M. A. Meyers (Dekker, New York, 1986)
p. 129.

4. P. Kasiraj, Shock Wave Consolidation of Metallic Powders (Ph. D. Thesis,
Caltech, 1984) p 84.

5. §.P. Marsh, ed., LASL Shock Hugoniot (University of California, Los Angeles,
1980) p. 221.

—88—

CHAPTER 6
TENSILE PROPERTIES OF A SHOCK CONSOLIDATED NI SUPERALLOY

6.1 Introduction

Considerable work involving rapid solidification processing (RSP) and
consolidation of Ni—based superalloys has been performed in the last several decades
[1,2,3]. Although the advantages of rapid solidification processing, in increased
homogeneity and refinement of microstructure have been demonstrated, many of
these advantages are lost when the consolidation of superalloy powders is
accomplished by conventional methods such as extrusion or hot—isostatic pressing.
These processes involve very prolonged thermal excursions during which the
temperatures approach the y' or 7" solvus. The intermetallic +" (Ni,Nb)
precipitates are the principal hardening phase. Consequently, the resulting
microstructure is coarser than that of the RSP powder. Furthermore, the
distribution of the y’and 7" phases is comparable to that developed in the material
produced by conventional casting techniques. Therefore, the advantages of rapid
solidification processing of Ni—based superalloys can be fully exploited only if the
powder is consolidated in the absence of prolonged heating.

Shock wave consolidation fulfills this requirement and can be used for
producing bulk solids with metastable structures. The consolidation of the powder
to full, solid density occurs by the passage of a shock wave of sufficient amplitude.
Within a short time interval (rise time of the shock wave ~80 ns), the energy of the
shock wave is utilized in plastically deforming the powder particles and reducing the

void volume to virtually zero. This results in preferential heating of the particle

—~89—

exteriors which can cause melting of particle surfaces and interparticle welding.
The heating and melting of the particle surfaces occurs in time durations too small
for atomic diffusion to occur over distances comparable to the particle size. The
melted interparticle material rapidly solidifies due to the heat flow towards
relatively cool particle interiors at rates as high as 101% ¢ /s [4], thereby retaining
the RSP microstructures.

In this investigation, a rapidly solidified Ni—based Pyromet 718 superalloy
powder (Nix ;Fesg 7Cty, 4Nbz gMos 9 Ti, gAlg ¢ wt%) was shock consolidated.
Mechanical (tensile) properties of the shock consolidated material were measured.
These were then compared with the properties of similar powder material
consolidated by hot—isostatic processing. Metallurgical observations were made to
elucidate the mechanical properties.

A comprehensive examination of the defect and precipitate structure of the
hot—isostatically pressed and shock—wave consolidated Pyromet 718 was conducted
using transmission electron microscopy by Prof. Naresh Thadhani [5]. Some of

these observations are presented.

6.2 Experiment

A rapidly solidified nickel—based superalloy powder (10-44 wm diameter)
obtained from Carpenter Technology was consolidated by a planar shock front of 9.5
GPa (stainless steel flyer impact velocity of 1.135 km/s). Details of the
consolidation technique are described elsewhere [6]. The individual powder particles
were microcrystalline with both cellular (0.25 — 0.5 um) and dendritic (1.0 — 3.0
ym) grains. A scanning electron micrograph of the as—received powder is shown in
Fig. 6.1a, and an optical micrograph of the polished and etched powder mounted in

lucite is shown in Fig. 6.1b.

—90—

The recovered consolidated samples (5 mm thick — 20 mm diameter) were
cut parallel and perpendicular to the shock direction. Optical and transmission
electron microscopy were performed to analyze the structural changes produced by
compaction at the interparticle regions and within particle interiors. For optical
microscopy, the sections were mechanically polished and etched with 5% FeCl,
solution. Samples for transmission electron microscopy were prepared by cutting
3.0 mm diameter discs from sections of the compact mechanically polished to 150
um thickness and final thinning by twin—jet electro—polishing with butyl cellusolve.

For tensile tests, small dog—bone shaped specimens (gage section 2.1 x 0.7 x
0.3 mm) shown in Fig. 6.2a were cut from the planar sections by electric discharge
machining. The specimens were pulled to failure in a three-legged frame designed
to minimize bending stress in the sample. The tensile sample is one of the three
load—bearing legs, shown schematically in Fig. 6.2b. Specimen extension was
measured using an LVDT referenced to the holes in the ends of the sample. Tensile
tests were conducted on the as—consolidated and heat treated samples of both the
hot—isostatically—pressed (hipped, obtained from Carpenter Technology) and shock
consolidated samples of Pyromet 718. The shock consolidated and hipped samples
were both subject to a standard heat treatment, 1066°C for one hour, followed by
initial aging at 740°C for 8 hours, and a final aging at 620°C for a period between 8
and 40 hours.

6.3 Results and Discussion

The rapidly solidified Pyromet 718 powder was dynamically consolidated to
full bulk density with a total calculated shock input energy of 400 J/g. The
microcrystalline powder particles are bonded together by interparticle regions which

appear to have melted and rapidly solidified to a fine microcrystalline structure.

—~9]—

Fig. 6.3a shows an optical micrograph of the shock consolidated Pyromet 718 alloy
(sectioned in the plane of the shock ). The powder particles have a dendritic
structure as observed in the RSP powder (Fig. 6.1), and are surrounded by a
featureless microcrystalline phase.

An optical micrograph of the Pyromet 718 alloy powder produced by
hot—isostatic pressing is shown in Fig. 6.3b. In the hipping technique, the powder
filled containers are placed in an autoclave which is subsequently heated and
pressurized normally to 1100—1200° C and 1 kbar, resulting in fully dense compacts.
As seen in Fig. 6.3b, a dark oxide layer (indicated by a glyceragia etch) surrounds

the particle surfaces.

Tensile Properties

Results from the tension tests, performed on the as—shock—consolidated and
hipped samples, revealed that the shock consolidated material had at least a 10%
higher yield strength (0.2% yield criterion) than the hipped material, although it
had practically zero ductility in the as—consolidated state. The hipped sample
showed up to 10% elongation and a higher ultimate tensile strength.

Following solution treating and aging, both the yield strength (YS) and
ultimate tensile strength (UTS) of the shock consolidated material showed a
significant increase over that of the hipped material. The tensile test results for
shock consolidated material and the hipped material are shown as plots of yield
strength vs. aging time (Fig. 6.4a) and ultimate tensile strength vs. aging time (Fig.
6.4b). Error bars in the figures show estimated experimental uncertainty based
primarily on irregularities in gauge section dimensions. It is clear from the two
plots that following heat treatment the properties of the shock consolidated material

are superior to those of the hipped alloy; UTS of the shocked alloy is 20% greater,

~92—

YS is 40% greater than the hipped alloy, and ductility is comparable. The tensile
properties measured in the hipped material are consistent with full—size tensile
samples of commercially produced superalloy 718 [7].

This improvement in the mechanical properties is attributed to
microstructural differences in the shock consolidated and hipped materials. As
indicated in the optical micrographs in Fig. 6.3a and 6.3b, the shock consolidated
sample has an extremely fine grain size in the interparticle regions created by the
rapid cooling of the interparticle melt formed during the passage of the shock wave.
In contrast, the hipped sample has a layer of oxidized material surrounding the

particles.

Transmission Electron Microscopy

In the process of shock consolidation, the energy of the shock is
inhomogeneously deposited at particle surfaces and within the particle interiors. As
discussed in Chapters 2 and 4, a non—negligible amount of energy is utilized in
plastic deformation of powder particle interiors. The remaining energy is deposited
near powder particle surfaces and results in local heating and melting.

The material melted during shock consolidation of Pyromet 718 rapidly
solidifies to a fine microcrystalline structure. Fig. 6.5 is a bright field TEM
micrograph of an interparticle region in shock consolidated Pyromet 718. The grain
size of the interparticle material is 0.05 — 0.1 yum, at least three times smaller than
the grain size in the particle interiors.

Evidence of extensive plastic deformation within powder particle interiors
was also observed. The deformation results in the formation of twins and
dislocation pile—ups and tangles within the individual grains. See Fig. 6.6

Shear localization bands, resulting from inhomogeneous plastic deformation

—~93—

during shock consolidation, are also observed within the particle interiors of the
Pyromet 718. Shear deformation is localized in narrow bands spanning over several
microcrystalline grains, but confined to individual powder particles. The shear
bands appear either as several isolated bands or as a packet of bands parallel and
nearly equidistant from one another. Fig. 6.7 is a TEM micrograph showing shear
bands in the shocked powder. These bands are 0.005 — 0.010 um thick and 1.0 pm
long in a microcrystalline (0.25 — 0.5 ym grain size) matrix. Contrast changes are
revealed as the bands cross grain boundaries. The material within the shear bands
reveals somewhat periodic fringe contrast, characteristic of micro—twinning.
Observations of fully heat treated samples of both the hipped and shock
consolidated samples showed considerably smaller precipitates present in larger
numbers in the particle interiors of shock consolidated material [5]. The higher

density of finer precipitates is expected to increase the strength of the material.

6.4 Conclusions

Shock consolidated Pyromet 718 superalloy powder shows mechanical
strength superior to the hipped material, and comparable ductility, after heat
treatment. This is attributed to improved interparticle bonding and grain size
refinement in interparticle regions of the shock alloy, and increased density of

precipitates.

Fig. 6.1 a) Scanning electron micrograph of as—received Pyromet 718 powder,

and b) optical micrograph of the powder after mounting, polishing, and etching.

—95—

Dog~-bone tensile fixture

O34 ya D3.2 mm
]r b+} 127mm ——--—___-|
p13 7
Ly
RO7—} |——
5.4 ©
0.4,
\P 102
2.1mm
Tensile
Specimen Va Pull~rod
meeN I 1
| 3 pe
Sh fo 1
Fig. 6.2 Illustration of the tensile sample and tensile frame. The sample is

scaled up to show detail.

Fig. 6.3 Optical micrographs of a) shock consolidated and b) hot isostatically

pressed samples of Pyromet 718.

~97—

_ 16+ a
S A
ht A A
= i 4
z é
z 12 ‘
oc
te
” '
rT A Shock comp.
> 0.8} @ hip comp.
oO 32 48
AGING TIME HRS
a 1.8;
= 1.6} A i 4
& i é
“ue ¢ ;
ts ¢
= 12; Ashock comp.
@ hip comp.
10# _. ,
0 16 32 48

AGING TIME HRS

Fig. 6.4 Plots of a) yield strength (YS) and b) ultimate tensile strength (UTS)

vs. aging time for shock consolidated and hot isostatically pressed Pyromet 718.

—98—

INTERPARTICLE

Fig. 6.5 Bright field TEM micrograph of an interparticle region of shock

consolidated Pyromet 718, revealing grain size decrease at particle interfaces.

—99—

Fig. 6.6 Bright field TEM micrograph of an intraparticle region of shock

consolidated Pyromet 718, revealing dislocation structures and twins.

—100—

Fig. 6.7 Bright field TEM micrograph of shear bands in shock consolidated

Pyromet 718. The bands cross several grains.

—101—

References

C. Hammond and J. Nutting, Metal Sci. 11, 474 (1977).

2. -D. F. Paulonis, J. M. Oblak, and D. S. Duvall, Trans. Am. Soc. Metals. 62,
611 (1969).

3. A. R. Cox, J. B. Moore, E. C. Van Reuth, Proc. 8rd Int. Symp. on Superalloys
(Claitors, Baton Rouge, 1976) p. 45.

D. G. Morris, J. Mater. Sci. 16, 457 (1982)
'N. N. Thadhani, A. H. Mutz, and T. Vreeland Jr., Acta Met. 37, 897 (1986).
N. N. Thadhani and T. Vreeland Jr., Acta Met. 34, 2323 (1986).

B. P. Bardels, ed., Metals Handbook Ninth Edition Vol. 1 (ASM, Metals Park
Ohio, 1978) p. 219.

Pte

—102—

CHAPTER 7
MULTIPLE CAVITY AND NEAR NET SHAPE CONSOLIDATION

7.1 Introduction

Multiple cavity experiments are usually done by encapsulating samples of
powder in a solid, rigid fixture, and subjecting the entire assembly to a flyer plate
impact or an explosive pressure pulse. The samples undergo equivalent pressure and
energy conditions (if the samples are themselves similar) but the shock history of
each is fairly complex; waves traveling through the target wrap around the capsules
and subject different portions of a cavity to widely varying pressures [1]. Extensive
numerical simulations have facilitated interpretation of resulting compacts but are
not easily performed for every compact and geometry [2]. Simplifying the shock
conditions facilitates interpretation of the experiments.

Striking only porous materials of similar impedance to the encapsulated
samples creates samples subject to nearly one—dimensional shock conditions if the
shock geometry is one—dimensional; the waves in the porous media do not ‘wrap
around’, but proceed at approximately the same velocity as the shock wave in the
sample. A similar line of reasoning led to the embedded off—center tube technique
used in a cylindrical geometry [3].

Using this design philosophy, three types of plane wave experiments have
been conducted: powders were loaded into cavities drilled into sintered bronze foam,
thin stainless steel dividers were used to separate three sectors of powders of

differing particle size distribution, and a pressed ’green’ compact of discontinuously

—103—-

reinforced metal matrix composite material was embedded in a ceramic powder of
similar shock impedance. Each of these target geometries was shocked by a flyer

plate impact. The specific geometries and procedures are somewhat different, but
the idea is consistently the same. The advantages in analytic experimentation are

paralleled by possible technological advantages.

7.2 Experiments
Porous sintered bronze fixture

Commercially made porous bronze filter material (Pacific Sintered Metals
F—100) was machined into 32mm dia. x 9.5mm thick disks with four 10 mm holes.
The sintered bronze has a mean density of 4.7 g/ cm® corresponding to a porosity of —
46%. (See Fig. 7.1.) The cavities were separately loaded with carbonyl Ni powders
of differing morphologies and pressed to 46% porosity. The carbonyl process nickel
powders, obtained from Inco Metals, were 7—10um spheres, 5.5—-6.5 ym spiked
spheres, 1 um thick x 30 ym flakes, and 2um thick x 10 um long filaments. Shot 25
was performed using the Keck Dynamic Compactor, a propellant—charged gas gun
accelerating a 5mm 303 SS flyer plate to 1.17 km/s. This impact shocked the Ni
powders to a calculated pressure of 9.1 GPa and deposited approximately 430 J/g of
thermal energy into the powder. Three well—bonded pills were recovered. A
micrograph of the compacted spiked spherical powder is shown in Fig. 7.2. The flaky
powder did not bond well, and was apparently contaminated by the roller lubricant

used in manufacturing.

Stainless steel dividers
We surmised that cavity partitions with thicknesses on the order of a particle

diameter or less would have minimal effect on the shock loading conditions of the

—104—

material in the fixture. If the materials themselves were nearly the same, the shock
conditions in each would not interfere with the others, thus each would be subject to
the same uniaxial shock conditions. The 304 stainless steel (SS) shim stock used
was 0.13 mm thick. The C350 maraging steel powder was sieved into various
fractions from 44—300 ym dia. The first cavity was loaded with 9.5g 125—150m
plus 9.5g 53—74um powder. The second cavity had 9.5g 180—300um and 9.5g
44—53um powder. The third cavity was filled with 19g of 180-300 wm powder. The
powders were consolidated by a 5mm thick 303 SS flyer plate accelerated to 1.28
km/s. The effect of particle size on energy distribution and bond quality is being
explored. The shot resulted in three sectors of seemingly well—bonded material (See
Fig. 7.3). Uniform consolidation is demonstrated by the lack of flyer plate
deformation, divider buckling, and retention of compact shape. Further
examination showed incomplete interparticle bonding throughout the compacts,

unrelated to the cavity dividers.

Zirconia powder shock transmitting media

A mixture of 10% Vol. 90 um irregular SiC powder (Electro Abrasives 180
grit) and balance 10 ym Ti 6Al 4V alloy (Powder Metals Inc.) was pressed to a
green shape shown in Fig. 7.4a under a static load of 150 MPa. The green compact
was sufficiently strong to retain shape during careful handling. It was placed flat
side up on a bed of —325 mesh zirconia powder (Cerac Z—1041), and more ZrO
powder was added to completely embed the green. The loaded target was covered
with a 0.13mm thick 304 SS cover plate to retain the powder and then evacuated
and impacted by a 5mm thick 303 SS plate at 1.0 km/s (Shot #59). The impact
was sufficient to densify and bond the compact but not the zirconia. The resulting

compact (Fig. 7.4b) is strained nearly uniaxially with 6% of uniform radial

—105—

expansion. The radial expansion is caused by an insufficient packing density of the
zirconia of only 47% (53% porosity), relative to the Ti + SiC, which is pressed to
59% of solid density.

7.3 Discussion

The three experiments described above each rely on a combination of rigid
radial confinement and uniform strain in the shock direction to retain a one
dimensional shock front. The effects of edge wave interactions at the rigid target
and flyer plate edges are therefore of critical interest. Determining the extent of
inter—cavity interactions is also important.

When a flyer plate strikes a porous media contained in a rigid fixture (and
not the fixture itself), a plane shock wave is generated in the media, as well as a
radial compression wave in the fixture at the powder—fixture interface. The
pressure of the wave in the fixture at the interface is determined by the shock
impedance match between the shocked powder and the fixture material and the
geometric boundary conditions. It may be a considerable fraction of the shock
pressure in the powder. Note that the pressure in the shocked powder is now being
released to an intermediate level. The wave in the cylinder attenuates in both
radial and longitudinal directions, and results in minimal (1%) expansion of the
fixture in our experiments. The three-element interactions of the flyer plate,
powder, and rigid cylinder have not been simulated in precisely the configuration
used; however, flyer plate experiments conducted using a metallic glass powder have
confirmed the retention of a well—defined, nearly planar consolidation front across
the entire compact surface as discussed in Chapter 2. This suggests the edge wave
dissipation is relatively minor in the geometry used.

The cavity to cavity interactions are more easily understood. Edge wave

—106—

interactions between neighboring shocked regions proceed at an angle determined by
Al’tshuler et al. [4]. Given a comparable shock speed and pressure, the angle is
quite small and the interaction minor. In the case of rigid dividers, the shock
impedance mismatch is large very close to the divider, but the shock state difference
will dissipate over a length of several divider thicknesses as the shock waves in the
divider ring down to the powder shock pressure.

The ceramic—composite shock impedance match affected the condition of the
MMC part fabricated. Matching the shock properties of a ceramic and metal
precisely proved quite difficult. We compromised by matching solid density and
shock speed as closely as possible. The resulting compact has not been mechanically
tested, but shows no signs of macroscopic cracking, buckling or bending. A careful
porosity match would result in a uniform strain in the shock direction. The shock

speed match prevents ‘wrap—around’ waves.

7.4 Conclusions

Three techniques have been demonstrated which maintain the plane wave
condition in the shock wave used to consolidate powders. They are useful for the
simultaneous consolidation of multiple samples. Valuable benefits of one
dimensional consolidation include nearly uniform processing conditions for the
compacts, and a simplified design and recovery of complex, near net shape parts
from a single cylindrical target design. These techniques are also useful in the study

of shock initiated chemical reactions in powder mixtures [5].

—107—

Fig. 7.1 Illustration of a four—cavity porous bronze target insert. The target

ring used is illustrated in Fig. 2.5. All dimensions are in mm.

—108—

Fig. 7.2 Optical micrograph of polished and etched shock consolidated spiky
spherical nickel powder, 5.5—-6.5 ym particle size.

—109—

Fig. 7.3 Photograph of recovered M350 maraging steel sectors of shot #63. The

three pieces did not bond strongly to the stainless steel divider sheets.

—110—

oe eee Oe

S rey

oS ?

Fig. 7.4 Photograph of a) the pressed green compact of Ti + SiC, and b) the

shock consolidated compact.

—111-—

References

1. G.E. Korth, J. E. Flinn, and R. C. Green, in Metallurgical
Applications of Shock—- Wave and High—Strain—Rate Phenomena, edited by L.
E. Murr, K. P. Staudhammer, and M. A. Meyers (Dekker, New York, 1986),
p. 129.

2. R.A. Berry and R. L. Williamson, in Metallurgical Applications of Shock
Wave and High Strain Rate Phenomena (Dekker, New York, 1986) p. 167.

3. P.S. Decarli and M. A. Meyers, Shock Waves and High Strain Rate
Phenomena in Metals, edited by M. A. Meyers and L. E. Murr (Plenum,
New York, 1981) p. 341.

4. LL. V. Altschuler, S. B. Kormer, M. I. Brazhnik, L. A. Vladimirov, M. P.
Speranskaya, and A. I. Funtikov, Sov. Phys. JETP. 11, 766 (1960).

5. B.R. Krueger and T. Vreeland Jr., in Metallurgical Applications of Shock
Wave and High Strain Rate Phenomena (Dekker, New York, 1991 in press).

—112-

CHAPTER 8
CONCLUSIONS AND REMAINING ISSUES

Several diverse and disparate systems of materials and experimental
techniques have been used to survey the effects of particle properties on energy
deposition during shock consolidation. These efforts have born fruit in several
quantitatively and qualitatively observed effects. The influence of particle size
distribution in the case of spherical powders has been well established in the
copper—constantan system. The influence of hardness in the mutual deformation of
hardened and unhardened maraging steel has been observed. The localization of
energy during the shock consolidation of a glass—forming alloy has been analyzed.
Several interesting developments in the consolidation of multiple samples with
uniaxial shock waves were made in the process of designing these experiments.
Observing the success of these designs led to the development of a useful technique
for consolidating a green compact to near net shape using a non—bonding pressure
transmitting media.

Placing all of these experiments, conducted over an extended time, in a
tightly organized framework (if possible) would create the illusion that a complete
understanding of the shock consolidation process at the particle scale has been
achieved, and the shock process rendered an accessible, reliable technology. This is
a bit off the mark. Shock consolidation as a manufacturing process remains an
uncertain proposition. In elemental metals and (most notably) in the nickel based
superalloy Pyromet 718, considerable success has been achieved in shock
consolidating well—bonded strong material. In seemingly similar systems, such as

M350 maraging steel, good consolidation has been very difficult. The metallurgical

—113-

root of this contrast is not clear.

On the other hand, several rays of light are visible. The Pyromet 718 not
only bonded well, but showed enhancement over conventionally processed material.
Post—shock annealing to remove residual stresses (and possible micro—cracks)
proved highly effective. Irregular powders seem to bond more regularly; minimizing
the non—deforming contact areas available during shock consolidation will aid
bonding. A ‘morning star’ shaped spiked sphere is likely to be a nearly optimal
shape for shock consolidation. A nickel powder produced using the carbonyl process
has this morphology, and was consolidated easily. The built—in asperities are very
highly deformed in consolidation. The use of a non—binding media, such as the
zirconia powder used to embed the titanium + silicon carbide, showed the benefit of
having a library of media which do not easily consolidate; it is likely to be a large
library! Extending this concept is a highly viable path for scale—up of shock
experiments.

This work leaves many issues open. Higher strain rate stress—strain
experiments are needed to create an accurate thermal and deformation model. The
relationship between particle shape and energy deposition is not resolved within a
single material; producing a metallic glass forming alloy in spherical and irregular
forms, and consolidating them in a multicavity planar shock target would facilitate
such an experimental series. The role of surface contamination in stifling bonding
has been noted [1], but quantitative analysis of the effect is not abundant. Given
sufficient research in these areas, shock wave consolidation may prove to be a
reliable manufacturing technique. It remains a very interesting method for exposing
material to the extreme of deformation rates and pressures, and observing the

unique results of these conditions.

—114—

References

1. TT. Vreeland Jr., P. Kasiraj, and T. J. Ahrens, in
Mat. Res. Soc. Symp. Proc. Vol 28 (North Holland, New York, 1984) p. 139.

—115-

APPENDIX A
HUGONIOT: A PROGRAM FOR CALCULATING SHOCK CONDITIONS

OPTION BASE 0

DIM MATL@6, 14), NAM$(14), XX(30), YY(30), VAR$(2)

CLS

REM HUGONIOT; ANDREW MUTZ, 1984-90

REM HUGONIOT INTERSECTION PROGRAM NOW WITH POWDER METAL CAPABILITY
REM

PRINT " HUGONIOT INTERSECTION"

PRINT “ PROGRAM TO DETERMINE THE IMPACT PARAMETERS IN A TARGET."

DATA 8.9394, 3.933, 1.5, 3.94, 1.98, 7.3e-3

DATA 8.902, 4.59, 1.44, 4.523, 1.73, 5.7e-3

DATA 16.6786, 3.293, 1.307, 3.41, 1.6, .002

DATA 19.224, 4.029, 1.237, 4.03, 1.54, 3.2e-3

DATA 7.896, 4.569, 1.49,4.51, 2.17, .006

DATA 7.85, 3.574, 1.92, 4.57, 2.17, 6.0E-3

DATA 2.7069, 5.386, 1.339, 5.33,2.0, .013

DATA 2.785, 5.328, 1.338, 5.33,2.0, 1.3E-2

DATA 2.833, 5.041, 1.4200, 5.04, 2.1, 1.37E-2

DATA 2.48,5.22, 1.32, 5.22, 2.12, .0149

DATA 10.206, 5.124, 1.233, 5.12, 1.7, 3.5E-3

DATA 2.15, 1.84, 1.71, 2, 1.6, .0962

DATA 4.34, 4.94, .95, 6.16, 1.3, .00607

DATA 1.146, 2.201, .716, 2.53, 1.0, .13

DATA Copper, Nickel, Tantalum, Tungsten, 304 Stainless

DATA 1018 Iron, 1100 Aluminum, 2024 Aluminum, 921-T Aluminum
DATA Aluminum - 3% Li, Molybdenum, Teflon, Titanium, Nylon

FORI = 1T0O 14

FOR j = 1TO6
READ MATLG, D)
NEXT j
NEXT I

LD = MATL(L, 14): LWS = MATL(2, 14): LCOEF = MATLG, 14) ’Nylon BACKING
FOR I = 1 TO 14: READ NAM$(1): NEXT I

REM J=1 -- DENSITY, J=2 -- WAVE SPEED, J=3 -- COEFFICIENT OF PARTICLE VEL.
REM J=4 -- SOUND SPEED, J=5 -- GRUNEISON’S COEF., J=6 -- ISEENTROPIC COMP.
Main: DO

PRINT "THE FOLLOWING MATERIALS’ PROPERTIES ARE BUILT IN:"
PRINT “1:Cu, 2:Ni, 3:Ta, 4:W, 5:304 SS, 6:1018 Fe, 7:1100 Al, 8:2024 Al,";
PRINT " 9: 921-T Al "

PRINT "10: 3% Li Al, 11: Mo, 12:Teflon, 13:Titanium, 14:Nylon"

PRINT “ Of the materials listed, powder data is available for all but";

~—116—

PRINT * 3,5,7, and 11."
PRINT "TO INPUT A MAT’L NOT LISTED, SPECIFY A "; 0; ", INSTEAD OF 1 - 13. "
PRINT “TO INPUT A MAT’L from file MAT.DAT, SPECIFY 15 INSTEAD OF 0 - 14. "
PRINT : PRINT
INPUT "TARGET MATERIAL: ",A
IF A = 0 THEN
INPUT " SPECIFY NAME OF TARGET MATERIAL : ", NAM$(0)
INPUT " DENSITY OF MATL (g/cm3) : ", MATL(1, 0)
INPUT * INTERCEPT OF SHOCK SPEED IN MATL (KM/S) : ", MATL(2, 0)
INPUT " COEF OF SHOCK SPEED : ", MATL(3, 0)
INPUT " GRUNEISON’S COEFFICIENT : ", MATL(5, 0)
INPUT "ISENTROPIC COMPRESSIBILITY (GPa-1) : ", MATL(6, 0)
END IF
IF A = 15 THEN
OPEN “mat.dat" FOR INPUT AS #1
FOR j = 1T0O6
INPUT #1, MATLG, 0)
NEXT j
INPUT #1, NAM$(0)
CLOSE #1
A=0
END IF
IF A <= 15 THEN
INPUT "Powder or Solid (P/S) : ", N$: M = 1
IF N$ = "P" OR N$ = "p" THEN M$ = "powdered" ELSE M$ = " solid"
END IF

LOOP WHILE A > 15

TNAME$ = NAM$(A): PRINT "Target mat’l is "; M$; " "; TNAME$: PRINT
TD = MATL(1, A): TWS = MATL(2, A): TCOEF = MATL(3, A)
TSS = MATL(4, A): TG = MATL(S, A): TISE = MATL(6, A)
IF N$ = "P" OR N$ = "p" THEN
INPUT "Distension of powder, = (final density/initial density) = ", M
END IF ;
FLYER: INPUT "FLYER MATERIAL: ", A
IF A > 14ORA < 0 THEN GOTO FLYER:
IF A = 0 THEN
INPUT " SPECIFY NAME OF FLYER MATERIAL : ", BNAM$
INPUT * DENSITY OF MATL (KG/M3) : ", FD
INPUT " INTERCEPT OF SHOCK SPEED IN MATL (KM/S) : ", FWS
INPUT " COEF OF SHOCK SPEED: ", FCOEF
INPUT " SPEED OF SOUND IN MATL (KM/S) :", FSS
ELSE
BNAM$ = NAM$(A)
PRINT "FLYER MAT’LIS "; BNAM$: PRINT
FD = MATL(1, A): FWS = MATL(2, A): FCOEF = MATL(3, A)
FSS = MATL(@4, A)
END IF

INPUT “Flyer thickness in mm is: ", L
VelInput: INPUT "FLYER VELOCITY UPON IMPACT (KM/S) : ", VELFLY: PRINT

*hugoniot calc. function definitions
DEF FNFLYHUG (U) = FD * (VELFLY - U) * (FWS + FCOEF * (VELFLY - U))

—117-—

DEF FNHUGINT (U)
hugint! = M *(1 + FNFLYHUG(U) * TISE) - 1
hugint2 = 1 + (1 + TG/2) * FNELYHUG(U) * TISE
FNHUGINT = U * 2 - FNFLYHUG(U) / TD * hugint1 / hugint2
END DEF

500 IF N$ = "s" OR N$ = "S" THEN

REM CALCULATE PARTICLE VELOCITY IN TARGET for solid-solid collision
A = FCOEF * FD - TCOEF * TD

IF A = 0 THEN
VELTAR = VELFLY / 2
ELSE

B = -FWS * FD - 2 * FCOEF * FD * VELFLY - TD * TWS
C = VELFLY * (FWS * FD + FCOEF * FD * VELFLY)
VELTAR = (-B - SQR(B*B-4*A*C))/2/A

END IF

SHOCKVELT = TWS + TCOEF * VELTAR

PRESS = VELTAR * SHOCKVELT * TD

ELSE ’Powder - Solid interaction

REM iterate for the intersection of the flyer and target hugoniots
Ul = VELFLY / 2: U2 = VELFLY * 5/6: Fi = FNHUGINT(U1): F2 = FNHUGINT(U2)
FOR I = 1 TO 500
ML = (F2 - Fl) / (U2 - U1)
UNEW = U2 - F2/ ML
IF ABS(UNEW - U2) / U2 < VELFLY / 10000 THEN EXIT FOR
U1 = U2: U2 = UNEW: Fl = F2: F2 = FNHUGINT(UNEW)
NEXT I
PRESS = FNFLYHUG(U2): VELTAR = U2
SHOCKVELT1 = M * 2 * PRESS / TD * (1 + (1 + TG / 2) * PRESS * TISE)
SHOCKVELT = SQR(SHOCKVELTI / (M * (1 + PRESS * TISE) - 1))
SHENERGY = PRESS * 1 / TD * (M - 1) /2 * 1000’KJ/KG
END IF

SHOCKVELF = PRESS / FD / (VELFLY - VELTAR)
a2 = FD * FCOEF - LD * LCOEF
IF a2 = 0 THEN a2 = 10* -6
B21 = FD * (FWS - 2 * (2 * VELTAR - VELFLY) * FCOEF)
B2 = B21 + LD *(LWS + 2 * VELFLY * LCOEF)
C21 = -FD * (2 * VELTAR - VELFLY) * (FWS - FCOEF * (2 * VELTAR - VELFLY))
C2 = C21-LD *(VELFLY *(LWS + VELFLY * LCOEF))
VFLY2 = (-B2 + SQR(B2 * 2-4 * a2 *C2))/2/a2
VF3INT = 2 * VFLY2 - (2 * VELTAR - VELFLY)
AI = TCOEF
BI = -TSS - 2 * TCOEF * VELTAR
CI = -PRESS / TD + VELTAR * (TSS + VELTAR * TCOEF)
VT2INT = (-BI - SQR(BI* 2-4 * AI *CD)/2/Al
TIMEB = L / (SHOCKVELF * FSS) * (FSS + SHOCKVELF + VELTAR - VELFLY)
A3 = FCOEF * FD - TCOEF * TD
IF A3 = 0 THEN
VTAR2 = VFLY2
ELSE

—118—

B31 = -FWS * FD - 2 * FCOEF * FD * VF3INT
B3 = B31-TD * TWS + 2 * TD * VT2INT * TCOEF
C31 = VF3INT * FD * (FWS + FCOEF * VF3INT)
C3 = C31 + TD * VT2INT * (TWS - TCOEF * VT2INT)
VTAR2 = (-B3 - SQR(B3 * B3 - 4 * A3 * C3)) /2/ AZ
END IF
SVELT2 = TWS + TCOEF * (VELTAR - VTAR2)
LSHOCKED1 = (SVELT2 + 2 * VELTAR) / (SVELT2 + VELTAR - SHOCKVELT)
LSHOCKED = LSHOCKED1 * TIMEB * SHOCKVELT / M
SVELF2 = FSS + (VFLY2 - VTAR2) * FCOEF
PRESS2 = FD * (VF3INT - VTAR2) * (FWS + FCOEF * (VF3INT - VTAR2))

Output Results
CLS
PRINT" TARGET “; M$; ""; TNAME$," FLYER "; BNAM$
IF N$ = "p" OR N$ = "P" THEN

PRINT "DISTENSION "; M, " “sn1"
END IF
PRINT "DENSITY "“; TD, “ "; FD
PRINT "WAVE SPEED "; TWS, " "; FWS
PRINT “SHOCK COEFF."; TCOEF, " "; FCOEF
IF N$ = "P" OR N$ = "p" THEN
PRINT "ISENTR. COM."; TISE, " "yee"
PRINT "GRUN. COEFF."; TG, " wy ten
END IF

PRINT : PRINT "FLYER VELOCITY ATIMPACT "; VELFLY; "KM/S"
PRINT "FLYER THICKNESS "; L; " mm.": PRINT

PRINT * PARTICLE VELOCITY IN THE TARGET "; VELTAR; "KM/S"
PRINT " SHOCK VELOCITY IN THE TARGET "; SHOCKVELT; "KM/S"

PRINT " PRESSURE OF THE SHOCK WAVE "; PRESS; "GPa"
PRINT " PRESSURE OF THE RELEASE WAVE "; PRESS2; "GPa"
IF N$ = "P" OR N$ = "p" THEN
PRINT " SHOCK ENERGY IN THE TARGET "; SHENERGY; “kJ/kG"

PRINT " RELEASE WAVE CATCHES SHOCK WAVE AT "; LSHOCKED;
PRINT "mm IN SHOCKED TARGET."
END IF
PRINT : PRINT
IF N$ = "P" OR N$ = "p" THEN
INPUT “Display the Time-Displacement diagram? (Y/N) :", YD$
IF YD$ = "Y" OR YD$ = "y" THEN GOSUB 1300 ELSE PRINT
END IF
IF N$ = "P" OR N$ = “p" THEN
INPUT "Display the Pressure - Particle Velocity diagram? (Y/N) :", YD$
IF YD$ = "Y" OR YD$ = "y" THEN GOSUB 1500 ELSE PRINT
END IF
INPUT " Calculate with same materials and new velocity? (Y/N) ", Y$
IF Y$ = "Y" OR Y$ = "y" THEN GOTO VelInput
INPUT * Calculate with new matls? (Y/N) ", Y$
IF Y$ = "Y" OR Y$ = "y" THEN
CLS
GOTO Main
END IF
END

—119~-

1300 HK DISPLACEMENT VS. TIME GRAPHICS 3 HEHE

’CALCULATE THE X-T PAIRS TO BE GRAPHED
XA = (VELFLY - SHOCKVELF) * L / SHOCKVELF: TA = L/ SHOCKVELF
TB = L/ SHOCKVELF / SVELF2 * (SVELF2 + SHOCKVELF + VELTAR - VELFLY)
XB = TB * VELTAR
XC1 = (SVELT2 + 2 * VELTAR) / (SVELT2 + VELTAR - SHOCKVELT)
XC = XC1 * TB * SHOCKVELT
TC = XC / SHOCKVELT
XD1 = 1/(1/(-SVELF2 + VFLY2) - 1 / VFLY2)
XD = XD1 *(TA-TB + XB/(-SVELF2 + VFLY2) - XA / VFLY2)
TID = (XD - XA) / VFLY2 + TA
XMIN = -INT(.5 *(L + VELFLY + 2.5)) *2
YMAX = INT((TID + TC) /2 + ABS(TID - TC)/2 + 1.5) + 1
YMIN = -1: XMAX = INT(.5 * (XC + 1.5)) *2
DLTX = INT((XMAX - XMIN) /8 + 1)
XMAX = XMAX + DLTX - (XMAX - XMIN) MOD DLTX: DLTPX = DLTX
IF YMAX < 3 THEN DLTY = .5 ELSE DLTY = INT((YMAX - YMIN) /6 + 1)
YMAX = YMAX + DLTY - (YMAX - YMIN) MOD DLTY: DLTPY = DLTY
TIT$ = "COLLISION OF FLYER WITH TARGET"
VAR$(0) = "DISTANCE (mm) ": VAR$(1) = "TIME (microsec)"
XX(1) = XD: XX(2) = XB: XX(3) = XC: XX(4) = 0: XX(5) = XA: XX(6) = XB
YY(1) = TID: YY(2) = TB: YY(3) = TC: YY(4) = 0: YY(5) = TA: YY(6) = TB
NNVAR = 2: NAX = 6: ITICK = 1: IDASH = 0

GOSUB 10000

GOSUB 10760
XX(1) = XD: XX(2) = XA: XX(3) = -L- VELFLY: YY(1) = TID: YY(2) = TA
YY(3) = -1: NAX = 3: IDASH = 20

GOSUB 10760
XX(1) = XB + VTAR2 * (TID - TB): XX(2) = XB: XX(3) = 0: XX(4) = -VELFLY
YY(1) = TID: YY(2) = TB: YY(3) = 0: YY(4) = -1
NAX = 4: IDASH = 20

GOSUB 10760
INPUT " ", NONS$
SCREEN 0: CLS
RETURN

1500 #4 PRESSURE VS. VELOCITY GRAPHICS "ebb

°CALCULATE THIRTY POINTS FOR THE POWDER COMPRESSION HUGONIOT
FOR I = 1 TO 30 ’PRESSURE = (I-1)/20 * PRESS

PR = (I- 1) / 20 * PRESS: YY(I) = PR

XX1 = PR/ TD *(M *(1 + PR * TISE) - 1)

XX(I) = SQR(XX1/(1 + (1 + TG/2) * PR * TISE))

NEXT I
’SPECIFY GRAPHICS PARAMETERS FOR LLPLOT

XMIN = 0: XMAX = INT(VELFLY + 1)

IF XMAX > 4 THEN DLTX = 1 ELSE DLTX = .5

YMIN = 0: YMAX = INT((PRESS * 1.5 + 1.5)): DLTY = INT((YMAX + 6) / 6)
YMAX = YMAX + DLTY - YMAX MOD DLTY

DLTPX = DLTX: DLTPY = DLTY: NNVAR = 2: NAX = 30: ITICK = 0: IDASH = 0
TIT$ = "COLLISION OF FLYER WITH TARGET"

VAR$(0) = "VELOCITY (km/s) ": VAR$(1) = "PRESSURE (GPa) "

GOSUB 10000

GOSUB 10760
°CALCULATE POINTS FOR FLYER HUGONIOT

FORI = 1TO9

—120—

PR = (I- 1) /6 * PRESS: YY(I) = PR
XX1 = (-FWS + SQR(FWS * 2 + 4 * PR/FD * FCOEPF)) / 2 / FCOEF
XX(1I) = VELFLY - XX1
NEXT I
NAX = 9
GOSUB 10760
*CALCULATE POINTS FOR RELEASE OF FLYER TO LEXAN BACKING
FORI = 1TO7
XX(1) = VFLY2 + (I- 1)/6 * (VELTAR - VFLY2)
YY1 = FD * (XX( - 2 * VELTAR + VELFLY)
YY() = YY1 * (FWS + FCOEF * (XX(I) - 2 * VELTAR + VELFLY))
NEXT I
NAX = 7
GOSUB 10760
*Calculate points for arrival of flyer release wave on compacted powder
*parametrize on velocity this time
FORI=1TOS5
VEL = (I- 1) * (VTAR2 - VFLY2) /4 + VFLY2: XX(1) = VEL
YY(I) = FD * (VF3INT - VEL) * (FWS + FCOEF * (VF3INT - VEL))
NEXT I
NAX = 5
GOSUB 10760
FOR I = 1TO7
VEL = (I- 1) * (VELTAR - VTAR2) /6 + VTAR2: XX(I) = VEL
YY(I) = TD * (VEL - VT2INT) * (TWS + TCOEF * (VEL - VT2INT))
NEXT I
NAX = 7
GOSUB 10760
*CALCULATE POINTS FOR LEXAN HUGONIOT
FORI = 1TO9
VEL = (I- 1) /6 * (VFLY2 - VELFLY) + VELFLY: XX(I) = VEL
YY(1) = LD * (VELFLY - VEL) * (LWS + LCOEF * (VELEFLY - VEL))
NEXT I
NAX = 9
GOSUB 10760
INPUT " ", NON$: CLS
RETURN

10000 ” ' SUBROUTINES “LLPLOT"

Log or Linear Diagram, Plotting, & Screen Printout Subroutines

’Nomenclature
.¢ Horizontal Axis Variable
*y Vertical Axis Variable

*The Following Variables Must be defined when the subroutines are called:
*DLTX & DLTY Scale intervals between printed values along the

, X & Y axes

*DLTPX & DLTPY Scale intervals between scale points along the

, X & Y axes

"*XMIN & XMAX Minimum & maximum values of the x axis variable
"YMIN & YMAX Minimum & maximum values of the Y axis variable

—121~—

*"NAX+1 No. of computed points of the diagram
*XX(NAX+1),YY(NAX+1) Arrays where computed points of the diagram
, are stored

*TIT$ String variable containing the diagram title
*VARS(NVAR) String variable containing the variable names
’NNVAR Dimension of the array VAR$

*VAR$(0) X axis variable name

*VARS(i) Y axis variable name, for i=1 to NNVAR
*ILOG =1 for a log (base 10) plot, YMAX & YMIN

, must be input as logs of the max and min Y’s

, and YY values are logs (LOG(YMAX)/LOG(10), etc)
*ITICK =1 for internal grid of ticks

*"IDASH =0 FOR Continuous line, 8 - 50 for dashed lines

owe

Diagram Frame & Axes Subroutine

Read Visualization Parameters & Plotting Data

NPX = 525 ’No. of pixels in the diagram along the X axis
NSX = 50 ’No. of pixels of the side edge

NPY = 280 ’No. of pixels in the diagram along the Y axis
NSY = 60 ’No. of pixels of the lower edge

SCREEN 9

*,..Draw figure frame & axes...

CLS : LINE (NSX, 350 - NSY - NPY)-(NSX, 350 - NSY)

LINE (NSX + NPX, 350 - NSY)-(NSX + NPX, 350 - NSY - NPY)
LINE (NSX, 350 - NSY)-(NPX + NSX, 350 - NSY)

LINE (NSX, 350 - NSY - NPY)-(NSX + NPX, 350 - NSY - NPY)

’,..Set unit points on the x axis...
THMAX = 5: JHMAX = 5
IMAX = INT((XMAX - XMIN) / DLTPX + .5)
JMAX = INT((YMAX - YMIN) / DLTPY + .5)
FOR I = 0 TO IMAX
IX = INTC * DLTPX * NPX / (XMAX - XMIN) + .5)
FOR JY = 1 TO 3: PSET (NSX + IX, 350 - NSY - JY): NEXT
NEXT
FOR j = 0 TO IMAX
JY = INTG * DLTPY * NPY / (YMAX - YMIN) + .5)
FOR IX = 1 TO'5: PSET (NSX + IX, 350 - NSY - JY): NEXT
NEXT
IMAX = INT((XMAX - XMIN) / DLTX + .5)
JMAX = INT((YMAX - YMIN) / DLTY + .5)
FOR I = 0 TO IMAX
IX = INT(I * DLTX * NPX / (KMAX - XMIN) + .5)
XVAL = XMIN + I * DLTX
YROW = INT((350 - NSY) / 8 + 2.5)
XCOL = INT((NSX + IX) / 8 - LEN(CHR$(ABS(XVAL))) /2 + .5)
LOCATE YROW, XCOL: PRINT XVAL;
NEXT

YVAL = RMNP / 10 *TO BE CHANGED FOR LOG AND LINEAR PLOTS
FOR j = 0 TO JMAX
JY = INTG * DLTY * NPY / (YMAX - YMIN) + .5)

—122—

IF ILOG = 1 THEN YVAL = 10 * YVAL ELSE YVAL = YMIN + j * DLTY
YROW = INT((350 - NSY - JY) /9 + 1): XCOL = 1
LOCATE YROW, XCOL
PRINT YVAL
NEXT
IMAX = INT((XMAX - XMIN) / DLTPX + .5)
JMAX = INT((YMAX - YMIN) / DLTPY + .5)
Ticks on the axes if itick < >0
IF ITICK < > 0 THEN
FOR j = 0 TO JMAX - 1
JY = INTG * DLTPY * NPY / (YMAX - YMIN) + .5)
FOR I = 0 TO IMAX - 1
FOR IH = 0 TO IHMAX
IX = INT (I + TH / (HMAX) * DLTPX * NPX / (XMAX - XMIN) + .5)
PSET (NSX + IX, 350 - NSY - JY)
NEXT
NEXT
NEXT
FOR I = 0 TO IMAX - I
IX = INT( * DLTPX * NPX / (XMAX - XMIN) + .5)
FOR j = 0 TOJMAX - 1
FOR JH = 0 TO JHMAX
JY = INT(G + JH / JHMAX) * DLTPY * NPY / (YMAX - YMIN) + .5)
PSET (NSX + IX, 350 - NSY - JY)
NEXT
NEXT
NEXT
END IF

FOR j = 0 TO JMAX - 1
JY = INTG * DLTPY * NPY / (YMAX - YMIN) + .5)
PSET (NSX + NPX, 350 - NSY - JY): DRAW "L8":
NEXT j
FOR I = 0 TO IMAX - 1
IX = INT(I * DLTPX * NPX / (XMAX - XMIN) + .5)
PSET (NSX + IX, 350 - NSY - NPY): DRAW "D3":
NEXT I
, 2 4 6 8 10
’...Print the title & variable names...
LOCATE 43, 10: PRINT TIT$;
YROW = INT((350 - NSY) / 8 + 3.5)
XCOL = INT((NSX + NPX * .9)/8 + .5)
LOCATE YROW, XCOL: PRINT VAR$(0);
FOR I = 1 TONNVAR
LOCATE INT((350 - NSY - NPY) /9 +3 +2*(-1)), 1
PRINT VARS(D;
NEXT
RETURN

10760 ’ * Plotting Subroutine **

*Can be called more than once to plot several curves in the same diagram

*Set points in the diagram

—123—

I=1
IXO = INT((XX(D - XMIN) * NPX / (XMAX - XMIN) + .5)
JYO = INT((YY() - YMIN) * NPY / (YMAX - YMIN) + .5)
FOR I = 2 TO NAX
IX = INT((XX(I) - XMIN) * NPX / (XMAX - XMIN) + .5)
TY = INT((YY(D - YMIN) * NPY / (YMAX - YMIN) + .5)
IF JY >= 0 AND JY <= NPY ANDJYO >= 0ANDJYO <= NPY THEN
LINE (NSX + IX, 350 - NSY - JY)-(NSX + IXO, 350 - NSY - JYO)
IF IDASH <> 0 THEN
XBEG = NSX + IX: YBEG = 350 - NSY - JY: XEND = NSX + IXO
YEND = 350 - NSY -JYO
GOSUB 11370
END IF

ELSEIF JYO >= 0 AND JY < 0 THEN

JYB = 0: IXB = IXO + INT((JYB - JYO) * (IX - IXO) / JY - JYO) + .5)

LINE (NSX + IXO, 350 - NSY - JYO)-(NSX + IXB, 350 - NSY - JYB)

IF IDASH <> 0 THEN
XBEG = NSX + IXO: YBEG
XEND = NSX + IXB: YEND
GOSUB 11370

END IF

350 - NSY - JYO
350 - NSY - JYB

ELSEIF JYO > NPY AND JY <= NPY THEN
JYB = NPY: IXB = IXO + INT((JYB - JYO) * (IX - IXO) / (IY - JYO) + .5)
LINE (NSX + IXB, 350 - NSY - JYB)-(NSX + IX, 350 - NSY - JY)
IF IDASH < > 0 THEN
XBEG = NSX + IXB: YBEG = 350 - NSY -JYB
XEND = NSX + IX: YEND = 350 - NSY - JY
GOSUB 11370
END IF

ELSEIF JYO < 0 ANDJY >= 0 THEN
JYB = 0: IXB = IXO + INT((JYB - JYO) * (IX - IXO) / (TY - YO) + .5)
LINE (NSX + IX, 350 - NSY - JY)-(NSX + IXB, 350 - NSY - JYB)
IF IDASH < > 0 THEN
XBEG = NSX + IX: YBEG = 350 - NSY - JY
XEND = NSX + IXB: YEND = 350 - NSY - JYB
GOSUB 11370
END IF

ELSEIF JYO <= NPY AND JY > NPY THEN
JYB = NPY: IXB = IXO + INT((JYB - JYO) * (IX - IXO) / (IY - JYO) + .5)
LINE (NSX + IXO, 350 - NSY - JYO)-(NSX + IXB, 350 - NSY - JYB)
IF IDASH <> 0 THEN
XBEG = NSX + IXO: YBEG = 350 - NSY - JYO
XEND = NSX + IXB: YEND = 350 - NSY -JYB
GOSUB 11370
END IF

ELSEIF JYO < 0 AND JY > NPY THEN
JYB = 0: IXB1 = IXO + INT((JYB - JYO) * (IX - XO) / JY - JYO) + .5)

—124—

JYB = NPY
IXB2 = IXO + INT((JYB - JYO) * (IX - IXO) / (JY - JYO) + .5)
LINE (NSX + IXB1, 350 - NSY)-(NSX + IXB2, 350 - NSY - NPY)
IF IDASH <> 0 THEN

XBEG = NSX + IXB1: YBEG = 350 - NSY

XEND = NSX + IXB2: YEND = 350 - NSY - NPY

GOSUB 11370
END IF

ELSEIF JYO > NPY AND JY < 0 THEN
JYB = NPY
IXB1 = IXO + INT((JYB - JYO) * (IX - IXO) / (JY - YO) + .5)
JYB = 0
IXB2 = IXO + INT(((JYB - JY) * (IX - IXO) / (JY - JYO) + .5))
LINE (NSX + IXB1, 350 - NSY - NPY)-(NSX + IXB2, 350 - NSY)
IF IDASH <> 0 THEN
XBEG = NSX + IXB1: YBEG = 350 - NSY - NPY
XEND = NSX + IXB2: YEND = 350 - NSY
GOSUB 11370
END IF
END IF
IXO = IX: JYO =JY
NEXT I
RETURN

, Plot data points as CROSSES subroutine *** He
11250’
FORI = 1 TO NAX
IX = INT((XX(D - XMIN) * NPX / (XMAX - XMIN) + .5)
JY = INT((YY() - YMIN) * NPY / (YMAX - YMIN) + .5)
IF XX() > = XMIN AND XX(I) < = XMAX THEN
IF YY) >= YMIN AND YY()) <= YMAX THEN
PSET (NSX + IX, 350 - NSY - JY): AAA$ = "R2L4R2U1D2": DRAW AAA$
END IF
END IF
NEXT I
RETURN

“eR Convert solid lines to dashed lines (from line routine)
11370 ” Dashes a line from (XBEG, YBEG) TO (XEND,YEND)
IF XBEG - XEND = 0 THEN
LLEN = 16/7 * ABS(YEND - YBEG)
XPROP = 0: YPROP = 7/ 16
ELSE
SLOPE = (YBEG - YEND) / (XBEG - XEND)
LLEN = SQR((XBEG - XEND) * 2 + ((YBEG - YEND) * 16 / 7) * 2)
XPROP = SQR(1 / (1 + SLOPE * 2 * (16 / 7) * 2))
IF SLOPE = 0 THEN YPROP = 0
IF SLOPE < > 0 THEN
YPROP = SQR(1 / (1 + (SLOPE * 16 / 7) * -2)) *7/ 16
END IF
END IF

—125-

IF XBEG > XEND THEN XPROP = -XPROP

IF YEND < YBEG THEN YPROP = -YPROP

FOR j = 1 TO LLEN STEP IDASH
CIRCLE (INT(XBEG + j * XPROP + .5), INT(YBEG + j * YPROP + .5)), 2,0
CIRCLE (INT(XBEG + j * XPROP + .5), INT(YBEG + j * YPROP + .5)), 1
PAINT (INT(XBEG + j * XPROP + .5), INT(YBEG + j * YPROP + .5)), 0

NEXT j

RETURN

—126—

APPENDIX B
THERCO: A PROGRAM FOR 1—D SIMULATION OF A THERMOCOUPLE

’A one dimensional shock wave thermal model simulation with two solid materials

*heat is deposited into a discretized system of masses w/connected interiors

* A. Mutz 10-2-90

DECLARE SUB TEXTPRINT (text$, X!, Y!, iflag%)

DECLARE SUB AXES (xmin!, xmax!, DLTX!, DLTPX!, ymin!, ymax!, DLTY!, DLTPY!, ITICK%,
JLOG %, TIT$, VAR$Q)

DECLARE SUB PLOT (iplot%, nax%, xmin!, xmax!, ymin!, ymax!, X!(), Y!Q)

DECLARE SUB MIX (bwidth, nax%, TC!())

DECLARE SUB TILT (bwidtht, nax%, TC!())

SCREEN 9

DEFINT I-J, M-N

DIM T(102), TC(2002), Ti(102), TP(102)" TEMPERATURE ARRAY

DIM xx(2003), yy(2003), a(2), B(2), VAR$(2)’GRAPHICS ARRAYS

PRINT "MATERIAL 1 NAME : "; “Copper"; " Material 2 Name :"; “Constantan"

x0$ = "75." ’Material 1 Thickness in microns

x1$ = "25." ’Material 2 Thickness in microns

x2$ = "185000." ’Shock energy in J/kg

x3$ = "20." initial temperature of material 1

x4$ = "20." initial temperature of material 2

x5$ = "8.9"

x6$ = "8.9"

TO = VAL(x3$)

Tl = VAL(x4$)

Energy = VAL(x2$)

ETHE = Energy / 1000

Roel = VAL(x5$) * 1000

ROES = VAL(x6$) * 1000

"L = 204 J/g, Cp = 0.35 J/gK, delTMelt = 588 K

delTMelt = 588

*CONVERT ALL TO STANDARD K,M,S,J UNITS, AFTER PRINTING

PRINT “Material 1 Thickness (um) ("; VAL(x0$); “) : "
10 = VAL(x0$) * .000001
PRINT "Material 2 Thickness (um) ("; VAL(x1$); "):"

rl = VAL(x1$) * .000001

x1$ = "1.11" ’Material 1 diffusivity cm2/s

x2$ = ".101 Material 2 thermal diffusivity"

x3$ = ".385" Material 1 conductivity kJ/msK

x4$ = ".035 "Material 2 conductivity

xS$ = "20."

x6$ = "1250."

PRINT “Material 1 THERMAL DIFFUSIVITY (CM2/S) ("; VAL(x1$); "): "
ALL = VAL(x1$) * .0001

PRINT "Material 2 THERMAL DIFFUSIVITY (CM2/S) ("; VAL(x2$); "):"
ALR = VAL(x2$) * .0001

—127—

PRINT “Material 1 THERMAL CONDUCTIVITY (KJ/MSK) ("; VAL(x3$); *) :"
Kl = VAL(x3$) * 1000
PRINT "Material 2 THERMAL CONDUCTIVITY (KJ/MSK) ("; VAL(x4$); "):"
KR = VAL(x4$) * 1000
TM = VAL(x6$)
nmax = 60
nmid = CINT(r0 / (r0 + rl) * nmax + 1)’THE FIRST NODE OF MATERIAL 2
RM = 0: rmil = 0: rm2 = 0
rpart = .000049 ’particle radius in material 0
rpartl = .00002
Vsh = 1500 ’ shock velocity in MKS
dr = (r0 + rl) /nmax’INCREMENT OF Thickness, WITH Fixed temp ON THE ENDS.
DT = .0000000005# ’TIME INCREMENT IN S
RS = ALR /dr*2* DT
LS = ALL/dr*2*DT
po = .5_ ’relative mass of outer shell to inner shell
fract = 1-(1/(1 + po)) * .33333333#? fraction of distance from
outer rad. to inner shell rad.
Lbeta = ALL / fract * 2 / rpart * 2 / po * DT /4
Lgamma = ALL / fract * 2 / rpart* 2 * DT /4
Rbeta = ALR / fract * 2 / rpart! * 2 / po * DT /4
Rgamma = ALR / fract * 2 / rpartl * 2 * DT / 4
ShWidth = rpart / Vsh ’rise time of shock wave in material
ShWidth1 = rpart! / Vsh
iwidth = CINT(ShWidth / DT) ‘number of time steps in shock rise time
iwidthl = CINT(ShWidth1 / DT)
ivel = CINT(1 / Vsh / DT * dr)’ number of distance steps per time step Sh Vel.
delt = Energy / KI * ALL * Roel * (po + 1) / po / iwidth
IF rpart > rpartl THEN ipart = CINT(rpart / dr) ELSE ipart = CINT(rpart1 / dr)

"initialize temperature profile-- **, ee, 3 HR Aetesetere eke

FOR i = 0 TO nmax

T(i) = 20
Ti(i) = 20° TEMPERATURES at t = 0
NEXT i

REM SET UP GRAPHICS

xmax = 1000000 * (rl + r0): ymin = 0: ymax = 1400: DLTY = 100: DLTPY = DLTY
nax = 60: NNVAR = 2: DLTX = xmax / 5: DLTPX = DLTX: xmin = 0:

TIT$ = "INITIAL CONDITION": VAR$(0) = "DEPTH": VAR$(1) = "TEMP": ITICK = 1
i$ = """ *pause for keystroke

a(1) = 1000000! * r0: a(2) = 1000000! * r0

B(1) = ymin + .9 * (ymax - ymin): B(2) = ymin

CALL AXES(xmin, xmax, DLTX, DLTPX, ymin, ymax, DLTY, DLTPY, ITICK, JLOG, TIT$, VAR$Q)
FOR ii = 1 TO nax: xx(ii) = dr * (ii - 1) * 1000000: NEXT ii
FOR c = DT TO .0000018 STEP DT’ TIME in 1 ns steps
ne = CINT(c / DT)

IFc <= (0 + rl + rpart) / Vsh THEN ’add heat
ShPosnF = Vsh *c

—128~

iShPosnF = CINT(ShPosnF / dr)
ShPosnR = Vsh * c - rpart
iShPosnR = CINT(ShPosnkR / dr)
OutC = .7
KinC = (1 - OutC)
FOR i = 0 TO nmid - ipart - 1
IF i > iShPosnR AND i < = iShPosnF THEN
TQ) = TQ) + delt * OutC
Ti(i) = Tidi) + delt * po * KinC
END IF
NEXT i
IF nmid - ipart < 0 THEN incr = nmid ELSE incr = ipart
FOR i = nmid - incr TO nmid - 1
IF i > iShPosnR ANDi <= iShPosnF THEN
Conc = -CSNG(i - nmid + ipart) / ipart / 2
T(i) = TQ) + delt * OutC * (1 + .5 * Conc)
Ti@i) = Ti(i) + delt * po * KinC * (1 + .5 * Conc)
END IF
NEXT i
IF ipart > nmax - nmid THEN incr = nmax - nmid ELSE incr = ipart
FOR i = nmid TO nmid + incr
IF i > iShPosnR AND i < = iShPosnF THEN
Conc = -CSNG(i - nmid - ipart) / ipart / 2
Ti) = Ti) + delt * OutC * (1 + .5 * Conc)
Ti(i) = Ti(i) + delt * po * KinC * (1 + .5 * Conc)
END IF
NEXT i
FOR i = nmid + incr + 1 TO nmax
IF i > iShPosnR AND i <= iShPosnF THEN
TQ) = T(i) + delt * OutC
Ti(i) = Ti(i) + delt * po * KinC
END IF
NEXT i

END IF

IF nc MOD 20 = 1 THEN
GOSUB 1440’ CALCULATE TOTAL ENERGY IN PARTICLE
LINE (xmin + .002 * (xmax - xmin), ymax * .998)-
(xmax * .998, ymin + .002 * (ymax - ymin)), 0, BF
LOCATE 4, 40: PRINT "Time "; c; : LOCATE 5, 40
PRINT "ECALC "; ETOT; : LOCATE 6, 40
PRINT "ETOT "; ETHE; : LOCATE 7, 40:
LOCATE 4, 20: PRINT "Cu": LOCATE 4, 60: PRINT "Co"
FOR ii = 1 TO nax: yy(ii) = T(ii - 1): NEXT ii
CALL PLOTtiplot, nax - 1, xmin, xmax, ymin, ymax, xxQ), yy()
CALL PLOT(O, 1, xmin, xmax, ymin, ymax, aQ), BQ)
END IF
WHILE i$ = ""
i$ = INKEY$
WEND
*Boundary conditions with half lumps on the ends and no flow
Told = T(0)
T(O) = Told + 2! * LS * (T(1) - Told) + Lhbeta * (Ti(O) - Told)’LEFT BC

—129—

Ti(O) = Ti(0) + Lgamma * (Told - Ti(0))

Told = T(nmax)

’ right be

T(amax) = Told + 2! * (T(nmax - 1) - Told) * RS + Rbeta * (Ti(nmax) - Told)
Ti(amax) = Ti(nmax) + Rgamma * (Told - Ti(nmax))

GOSUB 960 °*DIFFEQ PUSHER
IF nc MOD 2 = 0 THEN
Tnmid = T(nmid) + .5 * (T(nmid) - T(nmid + 1)
Tnml = T(nmid - 1) + .5 * (T(nmid - 1) - T(nmid - 2))
TC(ne / 2) = (Tnml + Tnmid) /2
END IF
NEXT c
bwidth = CINT(.5 * iwidth)
bwidtht = CINT(.75 * iwidth)

DLTX = .1: DLTY = 200: DLTPX = DLTX: DLTPY = DLTY: xmin = 0:

xmax = 1: ymin = 0: ymax = 2000

nax = 1800: NNVAR = 2

TIT$ = "COUPLE TEMPERATURE VS. TIME": VAR$(O) = "TIME": VAR$(1) = "TEMP": ITICK = 1

xmin = 0: xmax = 1.8
ymin = 0: ymax = 1400
CALL MIX(bwidth, nax, TCQ)) ‘convolve TC
CALL TILT(bwidtht, nax, TCQ) "Convolve TC
OPEN "“tcsil09.dat" FOR OUTPUT AS #1
FOR ii = 1 TO nax
yy(@ii) = TC(ii - 1)
Xx(il) = (ai - 1) / 1000
WRITE #1, .065 * (yy(ii) - 20), xx(ii) * 1000 + 420 ’EMF, Time in ns
NEXT ii
CALL AXES(xmin, xmax, DLTX, DLTPX, ymin, ymax, DLTY, DLTPY, ITICK, JLOG, TIT$, VAR$Q)
CALL PLOT(iplot, nax - 1, xmin, xmax, ymin, ymax, xx(), yy(Q)
CLOSE

STOP

960 REM DIFFEQ PUSHER,

*TSM = TEMP EXTRAPOLATION OF Material 0 NEAR Interface
*TLM = TEMP EXTRAPOLATION OF Material 1 NEAR Interface
*LSLOP= DT/Dx ON Material 0 SIDE OF INTERFACE

*RSLOP= DT/Dx ON Material 1 SIDE OF INTERFACE

FOR N = I TO nmax
IF T(N) > TM THEN
TP(N) = TP(N) + T(N) - TM

T(N) = TM
IF TP(N) > delTMelt THEN STOP
END IF
NEXT N
Told = T(0) *simplIFY calc.

FOR N = 1 TO omid- 1
TNEW = T(N) + LS * (T(N - 1) -2 * TIN) + T(N + 1)) + Lheta * (Ti(N) - TIN))

—130—

IF TP(N) > 0 THEN
TP(N) = TP(N) + TNEW - T(N)
IF TP(N) < 0 THEN
TNEW = TNEW - TP(N)
TP(N) = 0
END IF
TNEW = T(N)
END IF
Ti(N) = Ti(N) + Lgamma * (T(N) - Ti(N))
IF Told = TO THEN RETURN

T(N - 1) = Told
Told = TNEW
NEXT N

"INTERFACE TEMPERATURE CHANGE
N = nmid-1
Told = T(N) + LS * (T(N - 1) - T(N)) - (T(N) - TIN + 1) /(1/2/LS + 1/2/RS)
Told = Told + Lbeta * (Ti(N) - T(N))
N = nmid
TNEW = T(N) + RS * (T(N + 1) - T(N)) - (TON) - TIN - 1) /(1/2/ LS + 1/2/RS)
TNEW = TNEW + Rbeta * (Ti(N) - T(N))
IF TP(N) > 0 THEN
TP(N) = TP(N) + TNEW - T(N)
IF TP(N) < 0 THEN
TNEW = TNEW - TP(N)
TP(N) = 0
END IF
TNEW = T(N)
END IF
Ti(N) = Ti(N) + Rgamma * (T(N) - Ti(N))
T(N - 1) = Told
Told = TNEW

FOR N = nmid + 1 TO nmax - |
TNEW = T(N) + RS * (T(N - 1) - 2 * T(N) + T(N + 1)) + Rbeta * (Ti(N) - T(N))
IF TP(N) > 0 THEN
TP(N) = TP(N) + TNEW - T(N)
IF TP(N) < 0 THEN
TNEW = TNEW - TP(N)
TP(N) = 0
END IF
TNEW = T(N)
END IF
Ti(N) = Ti(N) + Rgamma * (T(N) - Ti(N))
T(N - 1) = Told
Told = TNEW
NEXT N
T(N - 1) = Told
RETURN

—131-

1440 ° **eK NUMERICAL INTEGRATION FOR ENERGY IN PARTICLE ; J/KG
ETOT1 = 0: ETOT2 = 0

ETOT1 = ETOT! + .5 * (T(0) - TO) + TP(ii)
ETOT1 = ETOT! + .5 * (Ti(0) - TO) / po

FOR ii = 1 TO nmid - 1
ETOT1 = ETOTI + (T(ii) - TO) + TP(ii)
ETOT1 = ETOT1 + (Ti(ii) - TO) / po

NEXT ii
Tmeanl = ETOTI /(.5 + nmid - 1) /(1 + 1/po)
ETOT1 = ETOT1 * KI / ALL/ Roel / 1000 / (.5 + nmid - 2) /(1 + 1/ po)

ETOT2 = ETOT2 + .S * (T(nmax) - TO) + TP(ii)
ETOT2 = ETOT2 + .5 * (Ti(nmax) - TO) / po

FOR ii = nmid TO nmax - 1
ETOT2 = ETOT2 + (T(ii) - TO) + TP(ii)
ETOT2 = ETOT2 + (Ti(ii) - TO) / po
NEXT ii
Tmean2 = ETOT2 / (nmax - nmid - .5)/ (1 + 1/ po)
ETOT2 = ETOT2 * KR / ALR/ ROES / 1000 / (nmax - nmid - .5) / (1 + 1/ po)
Tmean = (Tmeanl * (nmid) + Tmean2 * (nmax - nmid)) / nmax
ETOT = (ETOT1 * nmid + ETOT2 * (nmax - nmid)) / nmax
RETURN

DEFINT I-K, M-N

Linear Diagram, Plotting, & Screen Printout Subprograms

"xX Horizontal Axis Variable

"Y Vertical Axis Variable

"DLTX & DLTY Scale intervals between printed values along the

, X & Y axes

*DLTPX & DLTPY Scale intervals between scale points along the

, X & Y axes

*XMIN & XMAX Minimum & maximum values of the x axis variable
"YMIN & YMAX Minimum & maximum values of the Y axis variable
*Nax+1 No. of computed points of the diagram

*x(NAX+1), Y(NAX +1) Arrays where computed points of the diagram

, are stored

’TIT$ String variable containing the diagram title

*VAR$(0) X axis variable name

*VARS(1) Y axis variable name, for i=1 to NNVAR

*JLOG =TRUE for a log (base 10) plot, YMAX & YMIN

, must be input as logs of the max and min Y’s

"ITICK =1 for internal grid of ticks

, and YY values are logs (LOG(YMAX)/LOG(10), etc)
"IPLOT =0 to connect data points with lines,

=] to plot cross at data point,

—132—

=2 to plot crosses and connect points.

**** Diagram Frame & Axes Subprogram ***
SUB AXES (xmin, xmax, DLTX, DLTPX, ymin, ymax, DLTY, DLTPY, ITICK, JLOG, TIT$, VAR$Q)
DIM text(700)
true = -1
*,..Draw figure frame & axeS..........cececeeeneeeees
CLS : SCREEN 9, , 0, 0: ’set screen up for 640x350 graphics, establish plot margins
xl = xmin - .12 * (xmax - xmin)
yl = ymin - .19 * (ymax - ymin)
xr = xmax + .09 * (xmax - xmin)
yu = ymax + .05 * (ymax - ymin)
WINDOW axl, yl)-(xr, yu)

* draw plot frame ............... eee ceceeee ee eeeeeees
LINE (xmax, ymax)-(xmin, ymin), , B

’,.. Tick Axes and Set unit points on the axes.......

delx = .01 * (xmax - xmin): ticlen = .01 * (ymax - ymin): iflag = 1

FOR X = xmin TO xmax + delx STEP DLTPX
IF ITICK = 1 THEN
LINE (X, ymin)-(X, ymax), , , &H5500’dotted vertical line
ELSE
LINE (X, ymin)-(X, ymin + ticlen): LINE (X, ymax)-(X, ymax - ticlen)’ticks
END IF
Label point with value if interval is dltx
IF ABS((X - xmin) / DLTX - CINT((X - xmin) / DLTX)) < .05 THEN
xval$ = STR$(FIX(X) + CINT(10000 * (X - FEX(X))) / 10000)
yposn = ymin - .02 * (yu - yl)
CALL TEXTPRINT(xval$, X, yposn, iflag)
END IF
NEXT X

dely = .01 * (ymax - ymin): ticlen = .01 * (xmax - xmin): iflag = 3

FOR Y = ymin TO ymax + dely STEP DLTPY
IF ITICK = 1 THEN
LINE (xmin, Y)-(xmax, Y), , , &H5500’dotted horizontal line
ELSE
LINE (xmin, Y)-(xmin + ticlen, Y): LINE (xmax, Y)-(xmax - ticlen, Y)’ticks
END IF
’ Label point with value if interval is dity
IF ABS((Y - ymin) / DLTY - CINT((Y - ymin) / DLTY)) < .05 THEN
IF JLOG = true THEN YVAL = 10* Y ELSE YVAL = Y
YVAL$ = STR$(FIX(YVAL) + CINT(10000 * (YVAL - FIX(YVAL))) / 10000)
xposn = xmin - .01 * (xmax - xmin)
CALL TEXTPRINT(YVALS$, xposn, Y, iflag)
END IF
NEXT Y

*TITLE PLOT AND AXES 1.0.0... cccccces eee eeeeseeaeeeeee en

xposn = (xmax + xmin) / 2: yposn = ymin - .11 * (ymax - ymin): iflag = 2
CALL TEXTPRINT(VAR$(0), xposn, yposn, iflag)’print x-axis name

yposn = yu - .12 * (yu - yl): xposn = xl: iflag = 0

CALL TEXTPRINT(VAR$(1), xposn, yposn, iflag)’print y-axis name

xposn = (xmax + xmin) / 2: yposn = yl: iflag = 4

CALL TEXTPRINT(TIT$, xposn, yposn, iflag)’print plot title

END SUB

DEFINT I-J, M-N

SUB MIX *
SUB MIX (bwidth, nax, TCQ)

DIM T(nax)

*Convolve thermocouple curve with triangle function to simulate blurring of
*thermocouple boundary by particle stacking. normalize triangle.

*boundary is 2 * iwidth1 in total width.

bwsqt = (bwidth + 1) * 2

FOR ii = -100 - bwidth TO nax - 100 - bwidth

TCT =0
FOR jj = -bwidth TO bwidth
IF ii + jj < 1 THEN
TCT = TCT + 20 * (bwidth + 1 - ABS(jj))
ELSEIF ii + jj > nax THEN
TCT = TCT + TC(nax) * (bwidth + 1! - ABS(j))
ELSE
TCT = TCT + TC(ii + jj) * (owidth + 1! - ABSGj))
END IF
NEXT jj
T(ii + 100 + bwidth) = TCT / bwsqt

NEXT ii

FOR ii = O TO nax
TC(i) = T(ii)

NEXT ii

END SUB

DEFINT I-K, M-N

SUB PLOT **
**#* Plotting Subprogram Ro

*Can be called more than once to plot several curves in the same diagram
*Connect points in the diagram
SUB PLOT (iplot, nax, xmin, xmax, ymin, ymax, X(), YQ) STATIC
, limit plot to plotting area
VIEW SCREEN (PMAP(xmin, 0), PMAP(ymin, 1))-(PMAP(xmax, 0), PMAP(ymax, 1))
First plot lines (if desired) then mark data points
IF iplot = 0 OR iplot = 2 THEN
FOR i = 1 TO nax
LINE (X(i), Y@))-(XGi + 1), YG + 1D)
NEXT i

—134—
END IF

IF iplot = 1 OR iplot = 2 THEN
FOR i = 1 TO nax + 1
PSET (X(i), Y(@))
AAA$ = "R2L4R2U2D4"
DRAW AAA$
NEXT i
END IF
VIEW
END SUB

DEFINT I-K, M-N

SUB TEXTPRINT

SUB TEXTPRINT (text$, X, Y, iflag) STATIC? >t otobse ok stotesesteste ttt tok ott kk
” subroutine to place text at an arbitrary location (x,y)

* The 25th line is used as workspace by this program.

DIM text(1400)

*"TEXT$ is up to 80 characters
(x,y) is in world coordinates if WINDOW has been used.
*IFLAG designates point on text rectangle used as reference.

=0: (x,y) is middle left side

=1: (x,y) is middle of top side.

=2 : (x,y) is middle of box.

=3 : (x,y) is middle of right side.
=4: (x,y) is middle of bottom side.
vehar = 14 ’pixels high on EGA

hchar = 8 ’ pixels long on EGA

*SCREEN 9, , 1, O’use if 256K of video memory is available (causes screen blink)
LOCATE 25, 1: PRINT text$;
GET (PMAP(0, 2), PMAP(24 * vchar, 3))-
(PMAP(hbchar * LEN(text$), 2), PMAP(25 * vchar - 1, 3)), text
PUT (PMAP(0, 2), PMAP(25 * vchar - 1, 3)), text
*screen 9,,0,0 *use if 256K of video memory is available (causes screen blink)
htext = PMAP(0, 3) - PMAP(vchar, 3)’get label height in world units
Itext = PMAP(hchar * LEN(text$), 2) - PMAP(O, 2)’get label length in w. units.
xp = X: yp = Y - htext
IF iflag > 0 THEN xp = xp - Itext /2 ’place box position according to iflag
IF iflag = 3 THEN xp = xp - Itext / 2
IF iflag <> 1 THEN yp = yp + htext /2
IF iflag = 4 THEN yp = yp + htext /2

PUT (xp, yp), text
END SUB

DEFINT I-J, M-N

SUB TILT
SUB TILT (bwidtht, nax, TC())
DIM T(nax)

—135—

*Convolve thermocouple curve with box function to simulate tilt of
*thermocouple boundary . normalize box.
*boundary is iwidth in total width.

bwsq = (2 * bwidtht + 1)

FOR ii = -100 - bwidtht TO nax - 100 - bwidtht
TCT =0
FOR jj = -bwidtht TO bwidtht
IF ii + jj < 1 THEN
TCT = TCT + 20
ELSEIF ii + jj > nax THEN
TCT = TCT + TC(nax)
ELSE
TCT = TCT + TC(ii + jj)
END IF
NEXT jj
TGi + 100 + bwidtht) = TCT / bwsq

NEXT ii

FOR ii = 0 TO nax
TC(ii) = T(ii)

NEXT ii

END SUB