1. Introduction
1.1 History of Explosives
Explosives have been studied since the 10th century A.D. when Asian alchemists
stumbled upon an explosive mixture they called saltpeter or nitre. For centuries
afterward, the mixture was solely used for fireworks or for signaling distant ships or
armies. The Asian secret began to spread westward to Arabic and European
civilizations where it was further studied and developed into military weapons. Later, in
the 13th century, in the course of his studies in alchemy, Berthold Schwarz [1]
discovered the explosive properties of gunpowder which he then applied to firearms.
The need for stronger explosives led to the invention of high explosives (the above
forms of explosives are known as low explosives). Since low explosives react relatively
slowly (referred to as deflagration reactions), they produce moderately low pressure and
are useful in pushing objects such as bullets or cannon balls through a gun barrel. The
first high explosive was discovered by Italian scientist Ascanio Sobrero [2] in 1846.
Sobrero had invented nitroglycerin but its pure form proved too difficult and dangerous
for practical purposes until Alfred Nobel [2], motivated by the loss of his brother in an
explosion of some test material, invented a safer form of the explosive, dynamite, in
1866. Nobel still used nitroglycerin but he made dynamite a mixture of the liquid
nitroglycerin and some absorbent substance, or "dope," giving it a solid form.
Today, there are many different types of high explosives. Examples include Cyclonite
or RDX, HMX, PETN and tetryl. In their pure form, these explosives are dangerously
volatile so modern explosives are generally a mixture, or composite, of pure explosive
and inert materials. An example of a composite explosive is Composition 4, or better
known as C-4. The chemical name for C-4 is Cyclotrimethylene-trinitramine (a.k.a.
1,3,5-Trinitro-1,3,5-triazacyclohexane) or simply RDX for Royal Demolitions Explosive
[3].
Two classes of explosives used in industry are high velocity and low velocity explosives.
Some common high velocity explosives, 4572 - 7620 m/s (15,000 - 25,000 ft/s), are
TNT, RDX, PETN, Composition B, Composition C4, Datasheet and Primacord. Medium
velocity explosives, 1525 – 4575 m/s (5000 - 15,000 ft/s), include Ammonium nitrate,
Ammonium perchlorate, Amatol, Nitroguonidine, Dynamite and diluted PETN.
This study will explore both classes of explosives but will focus on medium velocity
explosives (composites) since they are safer and can be designed in such a way as to
provide idealized parameters for the job at hand.
1
1.2 History of Explosive Welding
Explosive welding (EXW) has been an industrial welding process since the late 1950s.
Since then, the process has been continuously refined and manufacturers worldwide
have explored the concept, determining the bonding parameters for nearly every
conceivable combination of metals. Since most of these companies are working on
their own funding, the work is proprietary in nature and information on the processes are
scarce outside of the company networks.
In process-controlled environments, explosives have been used to bond various metals
to each other. For example, when the United States mint stopped making coins of
~90% silver, sandwich or clad coins were made instead where a copper-nickel material
was bonded to pure copper.
This particular cladded coin was manufactured by the explosive bonding of large slabs,
which were ultimately rolled down to the required thickness. Initially, two slabs were
placed parallel to each other and approximately 6.4mm (0.25in) apart. An explosive
material was then placed on the top slab, and its detonation drove the slabs together
with enough force so that they became welded.
In other parts of the industry, EXW has been used to join stainless steel to ordinary
steel and controlled explosions have also been used on carbon to produce industrialtype diamonds used for grinding and polishing. A few examples of metals commonly
joined together using EXW techniques are summarized in Table 1.
2
X Carbon Steel
X X Stainless Steel
X X X Aluminum Alloys
X X X X Copper Alloys
X X X X X Nickel Alloys
X X X X X X Titanium
X X X X Silver
X
X X
X X X X X Gold
Platinum
X
X
X X
X
Carbon Steel
Stainless Steel
Aluminum Alloys
Copper Alloys
Nickel Alloys
Titanium
Silver
Gold
Platinum
Magnesium
X Magnesium
Table 1: Metals Commonly Joined Using the Explosive Welding Process [4]
2. Explosive Welding
2.1 Fundamentals
EXW is a solid-state metal-joining process that uses the enormous pressure-force
generated from an explosive. An electron-sharing metallurgical bond is created
between two metal elements [4]. Although very high temperatures result from the
explosion, the process occurs so quickly that there is insufficient time for heat transfer to
increase the temperature of the metals. As a result, EXW products do not have many of
the metallurgical characteristics of traditional welding, brazing or hot-rolled or forged
products may have. Some noteworthy differences between traditional welding
techniques and EXW are:






No heat-affected zones.
No continuous melt-bands with mixed chemical composition.
Minimal diffusion of alloying elements between components.
Product metals remain in wrought state so there are no continuous state
structures created and the tribological, mechanical and corrosion
properties are only minimally altered from their pre-bond conditions.
An effective joining method for nearly any metal combination (Table 1)
including combinations that are only achievable using the EXW process.
Well suited to metals that are prone to brittle joints when heat welded
(metals include aluminum on steel, titanium on steel).
Typically the process is completed at room temperature in air. It is also possible to
perform EXW in water or in a vacuum to minimize the high noise caused from the
explosion. In metal joining, if two materials can be brought close enough together, they
will bond at the molecular level. This normally does not happen because surface
contaminants prevent a close approach of surfaces. Normal welding overcomes this
problem by melting the materials so that they mix in liquid phases. During the explosive
process, surface contaminants are blown off the contact surfaces allowing virgin metal
to come into contact. Because the process occurs under pressures that are typically
measured in the GPa (millions of psi) the process is not well suited for brittle metals with
<5% tensile elongation or metals with a Charpy V-notch value < 13.5 N-m (10 ft-lb).
There are three bond types possible and each is dependent upon the parameters and
the type of set up used. The bond types are straight, direct metal-to-metal (DMM) and
wavy. DMM is the ideal bond type but it is difficult to achieve. Wavy bonds (Figure 1)
tend to be the strongest bonds and straight bonds (occur when the collision velocity is
too high) tend to be weaker. In section 3.5, the mechanics of the bonds are discussed
in greater detail.
3
Figure 1: Typical Wavy Bond Pattern of
an EXW Bond Interface (20X) [4]
2.2 Process Set Up
As previously stated, there are several configurations for the EXW process. Two
widely-known set ups are the parallel- and angle-bond geometries. Angle-bond plate
geometry is shown in Figure 2 where the standoff distance is non-uniform. As a result,
an angle (), measured from horizontal, represents the included angle between the two
plates. Another set up is known as the symmetrical oblique impact welding process.
Here, two plates are offset by an angle 2 in the shape of a “V.” Explosives line the
outer portions of each V and upon detonation, both halves are thrown against one
another.
Variations of these set ups have been used but the most commonly used process is the
parallel-bond geometry. This study will assess only the parallel plate geometry so will
be eliminated in the problem formulation.
2.3 Process Terminology [5]






Cladding Metal (or cladder) – The plate that is in contact with the explosive. It is
typically the thinner of the two metal components.
Base Metal – The plate that the cladding metal is bonded to.
Standoff Distance – The parallel separation distance between the cladding metal
and the base metal prior to the bonding operation.
Detonator, Booster and Explosive – The detonator and booster provide a
medium-strength explosion that initiates the detonation of the high explosive.
The high explosive provides energy for the forming process.
Assembly Operation – The process where the metals and explosive load are
placed into the proper positions for bonding.
Bonding Operation – The period in which the explosive detonation occurs and in
which bonding occurs. Duration of operation is measured in microseconds.
4
2.4 Description of Process
The assembly operation for the angle-bond process [4] is shown in Figure 2. At the
bottom of the stack, the base metal is held firm so that the cladding material can be set
at the appropriate standoff distance. Once the cladder is in place, the explosive
material is laid on top. After all the parts are in place and secured, the detonator is set
in the booster and explosive.
Detonator
Booster
Explosive
Cladder
α
Sacrificial Support
Standoff Distance
Base Metal
Figure 2: Parallel Bond Geometry Used for the
Explosive Welding Process
Once the explosion commences, there is no stopping the bonding operation for finetuning as can be done during traditional welding operations. Thus, the assembly
operation must be held to very strict tolerances. From initiation to completion, the
process is over in microseconds. Figure 3 shows what the bonding operation looks like
after detonation and before the operation is complete. The control volume is the region
where analyses in this study will be limited.
Expanding Gases
Detonation Front
Control Volume
Explosive
Cladder
Bonding Interface
Base Metal
Figure 3: Bonding Operation for the Parallel-Plate
Explosive Welding Process
5
2.5 Matallurgical Effects of Shock Waves in Metals
Pressure of the shock wave is the most influential parameter when discussing the
dislocation substructures generated by shock loading. As the pressure increases, so do
the dislocation densities of the material. Shorter pressure pulse duration allows less
time for dislocation reorganization within the material. So the substructures tend to be
more irregular since there is insufficient time for the dislocations generated by the peak
pressure in the shock front to equilibrate (Figure 4). Conversely, the cell walls become
better defined as the pulse duration increases as there is more time for dislocation
reorganization [10].
Figure 4: Effects of Pressure and Pulse Duration
on the Shock-Wave Response of Nickel [10]
During the EXW process, the explosion travels as a geometrical demarcation, or shock
front. In crystalline materials, a shock wave propagating through a material creates
lattice defects in the microstructure of the material [14]. The advancing shock front
leaves defects in the material, including linear dislocation arrays shown in Figure 5. In
this representation of the EXW process, peak pressure of the shock wave is shown as a
simple, plane-wave shock.
When the shock wave propagates, it does so at speed Vs, as expressed in equation (2)
of the next section. At this velocity, the pressure wave becomes the main driver of the
6
plastic deformation phenomena. As a result, the associated plastic deformation takes
on a form characteristic to shock wave microstructures. The propagation of a shock
wave in metals and alloys is represented below in Figure 5.
Pulse
Shock Front
Pressure
Rarefaction
Time
Figure 5: Idealized Shock Pulse Traveling
Through a Solid Metal or Alloy
The deformation induced in metals and alloys by this type of pressure pulse can be
separated into three regions: shock front, pulse and rarefaction or relief region. No
volumetric work is done in the pulse region since dV = 0 so compression of the solid
occurs in the shock front and relief regions only. These two regions make up the major
contributions to the shock deformation that produces permanent, residual
microstructural phenomena.
In Figure 6, a very simplified model represent the progress of a shock wave through a
metal or alloy. Initially, the lattice structure is cubic but as the wave penetrates the
material, high deviatoric stresses distort the structure into a monoclinic lattice. If the
stresses reach a critical, threshold level, homogeneous dislocation nucleation can occur.
The mechanism of nucleation at the shock front is unique from homogeneous
nucleation in conventional deformation processes because in shock loading, the
dislocation interface separates two lattices with different parameters.
In frame (a) of Figure 6, the lattice structure is shown to be cubic and as the wave front
propagates through the material, the lattice structure is altered. In (b), the wave front is
shown to coincide with the first dislocation interface where the density of dislocations
depends on the difference in specific volume between the two lattices. Next, the front is
seen moving ahead of the interface in (c) and in (d), the deviatoric stresses build up
again as other layers continue to be formed.
Recent experiments show that the rarefaction region of the wave does not significantly
impact the dislocation generation since this portion of the wave enters into a material
that is already highly dislocated. As the material is repeatedly shock-loaded, the
increase in dislocation density is significantly reduced for the succeeding events
whereas the shock wave passing through the highly dislocated material is not much of
an effective dislocation generator.
7
In addition to elastic deformation, two other deformation mechanisms can be observed
in metals during a deformation process driven by an external load: Crystallographic slip
along distinct slip systems (crystal plasticity) and mechanical twinning [15]. Both
mechanisms provide shear deformation on distinct crystallographic planes but twinning
shear is defined to be a homogeneous shear, which restores the lattice in a new
orientation. Its magnitude is given by the crystallographic elements describing the
atomic movement and the orientation relationships between the twinned and untwinned
regions.
Figure 6: Progress of a Shock Front
in Metals and Alloys [10]
Twin lamellae are often associated with very narrow regions within deformation twins.
The total deformation produced by twinning is then given by the thickness of the twin
lamellae (or twin bands) and their separation (see Figure 7). The most important
consideration for deformation twins is to note that twinning is a highly favored
8
deformation mode under shock loading. In shock loading, it is possible to force metals
that do not typically twin by conventional deformation at ambient temperatures to twin.
Figure 7: (Left) Deformation Twins (Photographed With Polarized
Light, 1500x) and Atomic Arrangement at the Twinning Plane (Right)
3. Problem Formulation
3.1 Parameters for Assembly Operation
Acceptable product quality can be assured by selecting parameters for the EXW
process. Manufacturers of EXW produces have determined the parameters for most
metal combinations by years of testing but this information is highly proprietary and not
readily available to the public. To analytically determine baseline parameters, an
approach has been documented in the ASM Handbook for welding, brazing and
soldering [4]. This approach has been developed using basic geometry, physics,
thermodynamics and wave propagation solutions. In some instances, it is necessary to
determine parameters solely using empirical data.
In Figure 2, three of the fundamental parameters (base metal thickness, cladder
thickness and standoff distance) are shown. Additional information of importance is the
type of material for each element and their material properties, the explosive’s
properties and finally the desired state of the EXW product (bond quality). Listed in
Appendix A are the parameters used when formulating EXW processes. In following
sections, it will be explained which of these parameters must be known up-front
depending on the situation and the desired outcome of the product.
9
Vd
Shock
Front
Cladder
Vs
Vf
Vp
β
Vc
Base Plate
Figure 8: Details of the Control Volume
For the Bonding Operation
Figure 8 represents the control volume shown in Figure 3. An equation for the dynamic
bend angle β has been empirically determined. Knowledge of the material properties
for the cladder and weld velocities can be used to give an initial reference for the
dynamic angle, βmin as seen in equation (1).
(1)
 min  C 0
1000 H v
Vc 2
where Hv is the Vickers Hardness value and ρ is the density (in kg/m3) of the cladder
material. The velocity of collision, Vc (in m/s), will be determined in the next section.
Equation (1) uses a constant, C0 that is equal to 0.6 when the surfaces of the plates to
have a high quality, pre-cleaned finish and a value of 1.2 if the plates are less perfectly
cleaned.
3.2 Parameters for Bonding Operation
When process parameters are properly balanced, the contact surfaces form a liquid jet
that starts at the point of impact and is directed away from the welded seam. In steady
state conditions, this jet (made up of surface oxides, absorbed gases and other
contaminants of the plates) is formed between the two materials being bonded. Since
the two joined surfaces are cleaned and brought together under high pressure, solidstate welding is possible. This analysis, derived from [6], ignores the parameter Vj,
velocity of jet, since omitting this velocity does not impact the results of the process
parameters.
There is also an established maximum velocity for welding; above this limit, the thermal
effects weaken the joint. Since most conventional explosives have a detonation velocity
that is above the desired value, composite explosives are generally developed for a
10
particular process. If the detonation velocity is too great, the ductile limit of the cladder
will be exceeded and material fracture may occur.
Many industrial companies have proprietary blends created with specific detonation
velocities that are used to produce their products. Table 2 lists the detonation velocities
for four common explosives. The analysis outlined in this report will evaluate common
composites and explosives, pure chemical explosives and idealized composite
explosives.
Table 2: Detonation Velocity for Selected Chemical Explosives [7]
Explosive
Nitroglycerin
Ammonium Nitrate
Trinitrotoluene
Royal Demolition Explosive
Common Name
Nitro
N/A
TNT
RDX, Cyclonite
Vd
m/s
6100
3400
6900
8040
Vd
fps
20,000
11,150
22,650
26,400
One way to establish a constraint for the detonation velocity is to compare it to the sonic
velocity, Vs. Sonic velocity is the speed of propagation of a pressure disturbance
though a material.
The longitudinal wave speed in an elastic solid is equal to the elastic modulus divided
by the density or
(2)
Vs 
E

where Vs is in m/s. Thus E, the elastic modulus for the cladder, must be in N/m 2 and ρ,
the density of the cladder, must be in kg/m3.
Empirical data has also determined that the wave propagation from the explosion
should not exceed the sonic velocity or else the pressure gradient in front of the shock
becomes too great and fracture of the cladder can occur. Keeping Vd within 100 - 120%
of the sonic velocity enables the shock front to broaden as it propagates, reducing the
pressure on the cladder.
Another empirical relationship has been developed for the density of the explosive. To
determine the density of the explosive as a function of the detonation velocity, the
relationship given in equation (3) can be use. This value becomes important when
calculating the explosive mass needed for the bonding operation and for calculating the
shock properties.
(3)
Vd = 1440 + (4.02)ρ0
where Vd is in m/s and ρ0 is in kg/m3.
11
90-½(β-α)
Vd
β-α
Vp
Vf
β
Vc
90-½(β-α)
90-½(β+α)
Figure 9: Geometry for Bonding Operation Velocity Vectors
There are several possibilities of allocating directions to the plate velocity, V p. In Figure
9, Vp is assumed to bisect the angle between the portion of the plate already
accelerated, behind the detonation front and the undeformed portion. This assumption
was justified in [10] for the case of an axisymmetric conical shell by considering the
continuity of mass flow through the collision point. Equations for the remaining
parameters of the bonding operation can be determined from the geometry shown in
Figure 9. Vector calculus, trigonometric functions and plane triangle formulas lead to
the following equations:
(4)
     Vp
Sin

 2  2Vd
Since Vd is known from Table 2, Vp can be determined by rearranging equation (4) into
the following
(5)

V p  2Vd Sin   ; since α = 0 for parallel plate bonding.
2
A similar approach can be used to determine the contact velocity V c.
(6a)
     Vc Sin
Cos
and (6b)

Vp
 2 
     Vc Sin
Cos

Vf
 2 
Since Vp is known from equation (5), and because α = 0,
(7)
Vc  V p
Cos / 2
 Vd  V f
Sin
where Vf is the velocity of the plate with respect to an observer moving with velocity V c.
12
During the evolution of EXW, parameter limits have been determined for the bonding
operation. Most of this information has been developed using trial and error. In
developing a process analytically, however, parameters can be limited to the
established ranges shown in Table 3.
Table 3: Estimated Ranges for Bonding Operation Parameters
Parameter
Vd
Limits
100% < Vs < 120%
0
140 < 0 < 900
m/s
kg/m3
Vp
250 < Vp < 500
m/s
Vc

d
1500 < Vc < 3500
5.0 <  < 20.0
if 0 < tc < 6.5, d = 2tc
if 6.5 < tc < 13, d = tc
m/s
deg.
mm
d
Units
mm
It has also been empirically determined that delamination occurs when d > 2tc, a wavy
bond occurs when tc ≤ d ≤ 1.5tc and a laminar bond occurs when d ≤ 0.5tc. Additional
considerations for the parameters include



(8)
Limiting Vc < Vs (Vc = Vd when α = 0)
Cladder ductility > 5% in tension to ensure β is feasible
Determining the explosive loading parameter, L
L
 y tc
d
2
where σy and ρ are, respectively, the yield stress and density for the cladder material.
Empirical data led to the derivation of Equation (8) so the units are meaningless. The
units used for L are mass per unit area (kg/m2).
3.3 Application of Thermochemistry
A thermochemical approach can be used to determine the energy released during the
bonding operation of Figure 8. When conventional explosives are used (nitroglycerin,
ammonium nitrate, TNT or RDX), parameters such as Vd and ρ0 are readily found in
most explosives reference manuals. However, if a particular process outcome is
desired, it is necessary to back-calculate the approach outlined in this study to
determine the values of the parameters listed above.
For chemical explosives, it is necessary to determine the energy release and
temperature from combustion in most cases. This is accomplished by balancing the
chemical reaction and determining the products after a catalyst is applied. Reference
manuals such as [7] and [8] list the chemical compositions (reactants) for several
explosives.
13
Four explosives and their chemical compositions are listed in Table 4. For the
explosives listed, it was necessary to determine the heat of formation (HoF) for both
reactants and products. Reference manuals typically list the HoF for the reactants and
the HoF for the products can be determined by adding the individual HoF for each of the
product’s components.
For example, the HoF for the addition of N20(g) and 2H2O(g) are 82.05 and 2(-241.8)
kJ/mol for a total value of -401.55 kJ/mol (values for the HoF for the products of Table 4
are listed in Appendix B).
Table 4: Heats of Formation For Combustion of Selected Conventional Explosives
Common
Name
Nitro
Explosive
Nitroglycerin
C 3 H5 N 3 O 9
HoF (R)
kJ/mol
-333.66
Reactants
Products
(Gaseous)
3CO2+2.5H2O+1.5N2+0.25O2
HoF (P)
kJ/mol
-1785.0
Ammonium Nitrate
N/A
NH4NO3
-365.14
N2O+2H2O
-401.55
Trinitrotoluene
TNT
C 7 H5 N 3 O 6
C 3 H6 N 6 O 6
227.00
6CO+2.5H2+C
3CO+3H2O+1.5N2
-670.80
Royal Demolition Explosive
RDX, Cyclonite
83.820
-1060.8
Once the heats of formation have been determined, it is then necessary to convert the
values into useful data for this analysis. For the problem at hand, the energy released
for each explosive is needed. To determine energy released,
(9)
∆E = ∆Hf(reactants) - ∆Hf(products)
Where ∆Hf( ) is the HoF. Table 5 is a summary of these values for the four explosives
listed above (values for ∆E > 0 represent an exothermic reaction). The final value for
∆E in kJ/kg is useful once the amount of explosive for the operation is known. By
multiplying the final values in Table 5 by the mass of the explosive, energy release for
the given problem can be calculated.
Table 5: Energy Release Values for Selected Conventional Explosives
Explosive
Nitroglycerin
Ammonium Nitrate
Trinitrotoluene
Royal Demolition Explosive
Common
Name
Nitro
N/A
TNT
RDX, Cyclonite
HoF (R)
kJ/mol
-333.66
-365.14
227.00
83.820
HoF (P)
kJ/mol
-1785.0
-401.55
-670.80
-1060.8
E
E
kJ/mol kJ/kg
1451.34 6393.6
36.41
160.4
897.80 3955.1
1144.62 5042.4
3.4 Shock Wave Analysis
Several methods can be used to model blast waves. One way to model the detonation
process is by a mathematical simulation of the Navier-Stokes Equations. In [11], the
governing equations are the Euler Equations for inviscid compressible flow with
chemical reaction added and are obtained from the compressible Navier-Stokes
equations. In [9], a code was developed to solve a generalized Langrangian analysis
14
for one dimensional particle velocity. The analysis assumes that the front of the
reactive wave acts as a non-reactive shock governed by a jump condition (Figure 11).
The laws of conservation of mass, momentum and energy can be used to form the
general equations of state for an inviscid flow of a non-conducting gas [13]. Typically,
this approach of mathematically determining the equation of state does not assume
continuity of the flow variables. These laws were derived as differential equations since
it was assumed that the flow is continuous. Equations (10) through (13) form the basis
of the numerical simulation using the Langrangian Analysis.
(10)
   1u 
2
  
  0 ; conservation of mass
 
  
 1

h
 t  h  0 h

t
(11)
1  
 u 
  
 
 t  h  0  h 
(12)
  
 v 
   P   0 ; conservation of energy
 t  h
 t  h
 1
 P 
   0 ; conservation of momentum
 h  t
where α is 1 for a symmetrical slab, 2 for a cylinder or 3 for a sphere (thus, α = 1 for this
analysis). Other parameters are t time, h Lagrangian coordinate, γ Eulerian coordinate,
ν specific volume, p pressure, ρ density and ε specific energy. The relationship
between the Eulerian and Lagrangian radius is given by
(13)
  
Vp    .
 t  h
The Lagrangian can be solved if the analysis assumes that the front of the reactive
wave is treated as a non-reactive shock governed by the Hugoniot jump condition of
Figure 11. In this study, a model of the explosion requires that the equations above are
applied to a flow region where the variables undergo a discontinuous change. A
discontinuity can be assumed in this case since there is a very large but finite gradient
in a region whose thickness tends to zero. The assumption for the existence of an
arbitrarily thin transition layer is used here because the dynamics for an inviscid, nonconducting gas assume that there are no characteristic lengths. These layers, in the
limit of vanishing thicknesses, are reduced to discontinuities. These discontinuities
represent shock waves.
In the approaches mentioned above, it is necessary to simplify the equations by
applying special conditions and assumptions such as steady state conditions and the
conservation laws for flow (conservations of mass, momentum and energy). Even with
these simplifications, solutions to these equations generally involve a high level of
program coding, processing power, time and funding. In this report, the shock wave
analysis will be for steady-state detonations as described in [11] and [12].
15
Represented in Figure 10, the following analysis considers a case where a block of high
explosive has initial pressure, specific volume and density Po, νo and ρo, respectively. A
shock wave travels at velocity Vd through the explosive media and at the wave front, the
shock compresses the explosive material to ν1 and raises the pressure to P1, initiating
the chemical reaction. At the rear of the reaction zone, the completed reaction gives a
pressure and volume of P2 and ν2. Since the reaction occurs within a region that is
generally between 200Å to 2mm thick, the pressure and volume are only considered at
the leading and trailing edges of the pressure wave.
Po, νo, Vo
P1, V1
Rear of Reaction
Po, νo, Vo
P1, V1
Rear of Reaction
Vd
Shock Front
Vd
Shock Front (Fixed)
Unexploded
Material
Vp
Vd-Vp
Unexploded
Material
Figure 10: Steady-State Conditions of the Detonation Process as Viewed by an
External Observer (Left) and by an Observer Traveling With the Shock Front (Right)
Similar to the approach taken in [9] and [10], here it is also necessary to apply the
appropriate assumptions and conditions to equations (10) through (13). Applying the
conservation laws for mass, momentum and energy to the system in Figure 10 gives
equations (14) through (16). The expansion of the reaction products follows the
Hugoniot curve of Figure 11. Points (Po, νo), (P1, ν1) and (P2, ν2) are collinear since the
velocities of the shock front and the rear of the reaction zone are equal under steadystate conditions. The collinear points lie on the Rayleigh line and are tangent to the
reaction products curve at the Chapman-Jouguet point (C-J point). Applying these
conditions to the Lagrangian equations above leads to the following relationships
(14)
Vd
v
 0
Vd  V p v1
2
Vd  Vp   P  P
Vd

1
0
v0
v1
2
(15)
(16)
 0  1 
1
Vd  V p 2  1 Vd 2  P1v1  P0 v0
2
2
16
P
Rayleigh Line
Reaction Products
o
Solid High
Explosives
(P2, v2)
(P1, v1)
C-J
o
o
(Po, vo)
v
Figure 11: Hugoniot Curves for the Detonation Process
The equations above can be rearranged to create equation (17) which represents the
change in pressure as a function of the change in specific volume.
2
(17)
P1  P0 
VV
Vd
v0  v1   d p
2
v0
v0
Another relationship for the detonation velocity can be derived using the thermodynamic
relationship
(18)
 P 
2
  2Vs    .
  
The thermodynamic relationship then leads to alternate expressions for V d, Vp
P1  P0
v0  v1
(19)
Vd  v0
(20)
Vp  v0  v1 
(21)
1   0 
P1  P0
v0  v1
1
P1  P0 v0  v1  ; the Hugoniot Equation.
2
Equations (17) and (19) through (21) assume that the specific volume behind the shock
front. This means that in equation (17), the vectors Vp and Vd are equal and that (P1 P0) can be used to solve the Hugoniot Equation since the equation of state for the
explosive reaction can be determined from the thermochemistry analysis.
17
Equation (2) gave a relationship for the sonic velocity, Vs. Since the sonic velocity is the
propagation velocity of the pressure wave behind the shock front, Vs can also be written
as
(18)
Vs = V d - V p
Using this relationship in equation (14) yields the following equation
(19)
Vs v1
 .
Vd v 0
3.5 Metallurgical Structure and Properties of Explosively Welded Joints
EXW products have a very characteristic and distinct bond profile. The most common
type of bond, previously shown in Figure 1 and below in Figure 12, is a bond where the
waves have a period (or wave spacing) designated by λ and a height associated with
the detonation velocity, Vd. Bonding is feasible because a jet, made of surface
containments and oxides, forms at the interface. Although this jet is very small, the
relative plastic deformation at the contacting surfaces is severe enough that the two
plates actually flow together to create the bond or weld zone. Inside this wavy zone are
some of the materials from the jet as well as some amount of fused metals. Most of the
metal is hardened by shock waves and there may also be anomalous slip and twinning,
increased dislocation density and some re-crystallization due to local heating [16].
Figure 12: (Left) Explosion Clad Plate Interface Of Zirconium and Steel,
(Right) Rolling up of Titanium Into Steel On the Top Of a Wave [5]
The wave formation in the figure above is referred to as the interfacial wave and the
criteria for producing this interfacial wave (introduced in section 2.1) is based on
qualitative studies. The basis for an analytical approach is the topic of much presentday research. Researchers are currently looking at the fluid mechanics of the collision
zone in order to establish a mechanism with a built-in means of explaining various types
18
of waves. In the zirconium-steel interface above, a high degree of rotation with little
appreciable melting is seen in the bond. However, in the image to the right, a large
vortex accompanies the wave and shows obvious phase changes at the vortex and at
the crest of the wave.
In order to model wave formation, it is important to realize that it is essentially a fluid
flow phenomenon. It is then possible to model the process by creating a system that
slows down the process in time and makes visual observation possible. High speed
photography has been effective in studies and has made observing the details of the
process possible. The liquid analogue shows that waves are caused by a combination
of the flow deformation components in front of and behind the stagnation point.
4. Problem Formulation
4.1 Problem Input
For this analysis, aluminum, titanium, nickel and steel were selected as the materials for
flyer plates. The goal of the analysis was to determine and compare the process
parameters from section 3 for several hypothetical EXW products. Listed in Appendix C
are the material properties and for the explosives, input data for the four analysis was
taken from Table 2. Two scenarios were selected for surface quality: 0.6 for highly
cleaned surfaces and 1.2 for less cleaned surfaces.
A program written in MATLAB (Appendix D) analyzed the input data from the above
sources and provided results for the minimum dynamic bend angle β, the desired
density of the explosive ρe, plate velocity Vp, pressure P and specific volume ν for the
various combination of materials and explosives.
19
4.2 Problem Results
Summarized in Table 6 are the results of the MATLAB program. Full program output is
given in Appendices E and F. Table 6 lists the data for highly cleaned surfaces (C 0 =
0.6) and Table 7 is for surfaces with a lower quality surface finish (C0 = 1.2). Grouped
together are the results for the four materials: Aluminum, titanium, nickel and steel. For
each material, the four explosives analyzed were nitroglycerin, ammonium nitrate, TNT
and RDX.
Table 6: Summarized MATLAB Results for EXW Parameters (C0 = 0.6)
0.6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Material
Aluminum
Aluminum
Aluminum
Aluminum
Titanium
Titanium
Titanium
Titanium
Nickel
Nickel
Nickel
Nickel
Steel
Steel
Steel
Steel
rho_m
2700
2700
2700
2700
4700
4700
4700
4700
8880
8880
8880
8880
7858
7858
7858
7858
Hv
70000
70000
70000
70000
60000
60000
60000
60000
75000
75000
75000
75000
155000
155000
155000
155000
E
7.00E+10
7.00E+10
7.00E+10
7.00E+10
1.10E+11
1.10E+11
1.10E+11
1.10E+11
2.07E+11
2.07E+11
2.07E+11
2.07E+11
2.05E+11
2.05E+11
2.05E+11
2.05E+11
Vd
3400
6100
6900
8040
3400
6100
6900
8040
3400
6100
6900
8040
3400
6100
6900
8040
beta
1.628
0.9074
0.802
0.6885
1.1424
0.6368
0.5629
0.4831
0.9292
0.5179
0.4579
0.393
1.42
0.7915
0.6997
0.6005
rho_e
487.56
1.16E+03
1.36E+03
1.64E+03
487.56
1.16E+03
1.36E+03
1.64E+03
487.56
1.16E+03
1.36E+03
1.64E+03
487.56
1.16E+03
1.36E+03
1.64E+03
Vp
96.6059
96.6082
96.6084
96.6086
67.7908
67.7916
67.7916
67.7917
55.1405
55.1409
55.1405
55.141
84.2655
84.267
84.2672
84.2673
P
1.60E+08
6.83E+08
9.05E+08
1.28E+09
1.12E+08
4.79E+08
6.35E+08
8.95E+08
9.14E+07
3.90E+08
5.17E+08
7.28E+08
1.40E+08
5.96E+08
7.90E+08
1.11E+09
v
0.002
8.49E-04
7.26E-04
6.02E-04
0.002
8.53E-04
7.29E-04
6.04E-04
0.002
8.55E-04
7.30E-04
6.05E-04
0.002
8.51E-04
7.27E-04
6.03E-04
Table 7: Summarized MATLAB Results for EXW Parameters (C0 = 1.2)
1.2
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Material
Aluminum
Aluminum
Aluminum
Aluminum
Titanium
Titanium
Titanium
Titanium
Nickel
Nickel
Nickel
Nickel
Steel
Steel
Steel
Steel
rho_m
2700
2700
2700
2700
4700
4700
4700
4700
8880
8880
8880
8880
7858
7858
7858
7858
Hv
70000
70000
70000
70000
60000
60000
60000
60000
75000
75000
75000
75000
155000
155000
155000
155000
E
7.00E+10
7.00E+10
7.00E+10
7.00E+10
1.10E+11
1.10E+11
1.10E+11
1.10E+11
2.07E+11
2.07E+11
2.07E+11
2.07E+11
2.05E+11
2.05E+11
2.05E+11
2.05E+11
Vd
3400
6100
6900
8040
3400
6100
6900
8040
3400
6100
6900
8040
3400
6100
6900
8040
20
beta
3.2561
1.8149
1.6044
1.3769
2.2848
1.2735
1.1259
0.9662
1.8584
1.0359
0.9158
0.7859
2.8401
1.583
1.3995
1.201
rho_e
487.562
1.16E+03
1.36E+03
1.64E+03
487.5622
1.16E+03
1.36E+03
1.64E+03
487.5622
1.16E+03
1.36E+03
1.64E+03
487.5622
1.16E+03
1.36E+03
1.64E+03
Vp
193.1924
193.21
193.212
193.2137
135.5749
135.5811
135.5817
135.5822
110.2774
110.2807
110.281
110.2813
168.5181
168.53
168.5312
168.5323
P
3.20E+08
1.37E+09
1.81E+09
2.55E+09
2.25E+08
9.59E+08
1.27E+09
1.79E+09
1.83E+08
7.80E+08
1.03E+09
1.46E+09
2.79E+08
1.19E+09
1.58E+09
2.22E+09
v
0.0019
8.35E-04
7.16E-04
5.94E-04
0.002
8.43E-04
7.22E-04
5.99E-04
0.002
8.47E-04
7.25E-04
6.01E-04
0.0019
8.39E-04
7.18E-04
5.96E-04
5. Conclusions
5.1 Discussion of Results and Summary
Figure 13 shows a plot of the results for the dynamic bend angle for aluminum, titanium,
nickel and steel as a function of the detonation velocity. In the equation for β, when the
material parameters are constant, the angle is then dependent upon only one variable,
Vc. Thus, the dynamic bend angle only changes by the inverse of the cladder velocity.
Once the materials for the product are chosen, prediction methods for β are then based
solely on the velocity of the cladder. And, as discussed earlier, Vc = Vd for scenarios
where α = 0.
Dynam. Bend Angle as Func. of Det. Vel. (Co = 1.2)
Dynam. Bend Angle as Func. of Det. Vel. (Co = 0.6)
3.5
1.8
Al
Al
Ti
Ni
1.4
St
1.2
1
0.8
0.6
0.4
Ni
2.5
St
2
1.5
1
0.5
0.2
0
3000
Ti
3
Min. Dynamic Bend Angle (deg)
Min. Dynamic Bend Angle (deg)
1.6
4000
5000
6000
7000
8000
9000
0
3000
4000
5000
6000
7000
8000
9000
Detonation Velocity (m /s)
Detonation Velocity (m /s)
Figure 13: Dynamic Bend Angles for Four Materials and
Four Types of Explosives. Material Surfaces Rated as
Highly Cleaned (C0 = 0.6) and Less Perfectly Clean (C0 = 1.2).
Similar trends are also noticed when comparing the results for the shock pressure
induced on the cladder material (Figure 14). Here, when material properties are held
constant, the only variable changing is then the detonation and associated plate
velocities, Vd and Vp respectively. The associated pressures are in the GPa range and
are orders of magnitude larger than the yield limits of the material. It is only because
the pressure is acted on the material on such short intervals is it possible to not fracture
the material. However, it is important to note the limits mentioned in section 3.2 since
exceeding the ductility of the cladder by more than 5% will lead to fracture of the
materials and the weld will be of poor quality.
21
Shock Pressure as Func. of Det. Vel. (Co = 0.6)
Shock Pressure as Func. of Det. Vel. (Co = 1.2)
1.40E+09
Ti
Ti
1.20E+09
Ni
2.50E+09
Ni
St
St
1.00E+09
Shock Pressure (Pa)
Shock Pressure (Pa)
Al
3.00E+09
Al
8.00E+08
6.00E+08
4.00E+08
2.00E+09
1.50E+09
1.00E+09
5.00E+08
2.00E+08
0.00E+00
3000
4000
5000
6000
7000
8000
9000
0.00E+00
3000
4000
5000
6000
7000
8000
9000
Detonation Velocity (m/s)
Detonation Velocity (m /s)
Figure 14: Shock Pressure Induced on the Material for Four
Different Materials and Four Types of Explosives. Material Surfaces
Rated as Highly Cleaned (C0 = 0.6) and Less Perfectly Clean (C0 = 1.2).
Since the calculated bend angle is actually the minimum angle, this value would be
used as a baseline when developing the process for an explosively welded product.
Once the baseline data is gathered from the analytical portion of the test, it would then
be necessary to gather actual test data. The entire process has not been modeled fully,
or at least models available to the general public, so test data is an absolutely critical
portion of the design phase.
Many companies have used similar analyses as shown in this study to begin their initial
development of an explosively welded product. Continuation of their research is
handled in highly proprietary manners. Testing of the products is dangerous and very
costly due to the difficulty in procurement of the explosive material. Additionally, it is
necessary to gain government approval and to meet safety and legal regulations prior to
research development. Because of these difficulties, the process is very limited to a few
companies in the United States but is more wide spread in countries such as Russia,
Germany and the United Kingdom where regulations tend to be lighter.
Despite recent advances in the filed, additional research is necessary in the explosive
welding field before the products gain more attention throughout the industrial
community. Additionally, it is important to continue educating consumers in order to
prove the concept viable in various industrial applications. Although EXW products are
an effective joining method for nearly any metal combination, the procurement of the
explosive material and the need for remote detonation locations pose significant hurdles
that must be dealt with before EXW products become an attractive business venture for
more companies.
22
APPENDIX A
Parameters Used for Calculations in the Assembly Operation
Description
Plate Angle, Lagrangian Constant
Clad Plate Area
Symbol

Ac
 min
C0
d
Hf
E
 0,  1

h
Hv
Dynamic Bend Angle
Surface Quality Constant
Standoff Distance
Heat of Formation
Elastic Modulus
Specific Energy
Eulerian Coordinate
Lagrangian Coordinate
Material Hardness (Vicker's)
Explosive Load
L
me
Mass of Explosive
Unit
deg., none
m2
deg.
None
mm
KJ/mol
GPa
kJ/kg
m
m
N/m2
kg/m2
kg
 0,  1
P0, P1
m3
Pa

0
kg/m3
kg/m3
y
tc
u
Vc
N/m2
mm
m/s
m/s
Detonation Velocity
Plate Velocity w.r.t. Vc
Vd
m/s
Vf
m/s
Cladder Plate Velocity
Vp
m/s
Sonic Velocity
Vs
m/s
Specific Volume
Explosive Pressure
Material Density
Explosive Density
Yield Strength
Cladder Thickness
Lagrangian Wave Speed
Collision (Weld) Velocity
23
APPENDIX B
Heat of Formation for Selected Molecules [8]
Molecule
(Compound)
Hof,
kJ/mol
H2(g)
0
O2(g)
0
N2(g)
0
H2O(g)
-241.8
CO2(g)
-393.5
N2O(g)
CO(g)
82.05
-111.8
24
APPENDIX C
Mechanical Properties for Select Materials (Adapted From [4])

kg/m3
2700
19320
4700
21450
1740
8880
8960
7858
7750
10490
Material
Aluminum
Gold
Ti
Platinum (Annealed)
Magnesium (Annealed)
Nickel (Annealed)
Copper (Annealed)
Steel (AISI 1022)
Stainless Steel (Custom Annealed)
Silver
25
Hv
2
N/m
70000
25000
60000
40000
40000
75000
50000
155000
292000
25000
E
GPa
70
77.2
110
171
44
207
110
205
200
76
y
MPa
28
140
100
59
33.3
360
375
APPENDIX D
MATLAB Code for Generating Process Parameters
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
EXPLOSIVE WELDING PROCESS
%%%
%%%
FLAT PLATE GEOMETRY (ALPHA = 0)
%%%
%%%
PROGRAM DETERMINES THE PARAMETERS NEEDED FOR THE WELDING PROCESS
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
clear;
clc;
disp('Enter data for the softer of the two materials being bonded');
disp(' ');
rho = input('Enter density (kg/m^3): ');
Hv = input('Enter Vickers Hardness Value (N/m^2): ');
E = input('Enter Modulus of Elasticity (N/m^2): ');
disp(' ');
Vd = input('Velocity of detonation for explosive used (m/s): ');
Co = input('Bonding surfaces: High quality pre-cleaned (0.6) or less perfectly cleaned (1.2)? ');
%
B_min_rad = Co*sqrt((1000*Hv)/(rho*Vd^2));
B_min_deg = B_min_rad*180/pi;
%
Vs = sqrt(E/rho);
%
rho_0 = (Vd-1440)/4.02;
%
Vp = 2*Vd*(sin(B_min_rad/2));
%
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
OUTPUT
%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
%
fid = fopen('parameters');
disp(' ');
disp('Min dynamic angle (deg): ')
disp(B_min_deg);
if B_min_deg <= 5
disp('**NOTE** Min dynamic angle is below recommended range')
elseif B_min_deg >= 20
disp('**NOTE** Min dynamic angle is above recommended range')
else
disp('Min dynamic angle is within recommended range')
end
%
disp(' ');
disp('Detonation velocity (m/s): ')
disp(Vd);
if Vd > 1.2*Vs
fprintf('**NOTE** Detonation velocity is above recommended range')
else
fprintf('Detonation velocity is within recommended range')
end
%
disp(' ');
disp(' ');
disp('Density of explosive (kg/m^3): ')
disp(rho_0);
if rho_0 <= 140
disp('**NOTE** Explosive density is below recommended range')
elseif rho_0 >= 900
disp('**NOTE** Explosive density is above recommended range')
else
disp('Explosive density is within recommended range')
end
%
disp(' ');
disp('Plate velocity (m/s): ')
26
disp(Vp);
if Vp <= 250
disp('**NOTE** Plate velocity is below recommended range')
elseif Vp >= 500
disp('**NOTE** Plate velocity is above recommended range')
else
disp('Plate velocity is within recommended range')
end
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
EXPLOSIVE WELDING PROCESS
%%%
%%%
FLAT PLATE GEOMETRY (ALPHA = 0)
%%%
%%%
THIS SECTION DETERMINES THE SHOCK PROPERTIES
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
P_0 = 101.325; % 1 ATM = 101.325 Pa
%
v_0 = 1/rho_0;
%
P_1 = Vd*Vp/v_0 + P_0;
%
v_1 = v_0 - (P_1 - P_0)*(v_0^2)/(Vd^2);
%
e = 0.5*(P_1 - P_0)*(v_0 - v_1);
%
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%
OUTPUT
%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
%
disp(' ');
disp('Pressure of shock (N/m^2): ')
disp(P_1);
%
disp(' ');
disp('Specific volume of explosive after detonation (m^3/kg): ')
disp(v_1);
%
27
APPENDIX E
MATLAB Program Output for Surface Condition, C0 = 0.6
1. Min dynamic angle (deg): 0.9074
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6100
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 1.1592e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 96.6082
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 6.8313e+08
Specific volume of explosive after detonation (m^3/kg): 8.4900e-04
2. Min dynamic angle (deg): 1.6280
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
3400
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 487.5622
Explosive density is within recommended range
Plate velocity (m/s): 96.6059
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.6014e+08
Specific volume of explosive after detonation (m^3/kg):
0.0020
3. Min dynamic angle (deg): 0.8022
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6900
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.3582e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 96.6084
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 9.0538e+08
Specific volume of explosive after detonation (m^3/kg): 7.2596e-04
4. Min dynamic angle (deg): 0.6885
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
8040
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.6418e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 96.6086
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.2752e+09
Specific volume of explosive after detonation (m^3/kg): 6.0177e-04
5. Min dynamic angle (deg): 0.6368
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6100
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.1592e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 67.7916
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 4.7936e+08
Specific volume of explosive after detonation (m^3/kg): 8.5307e-04
6. Min dynamic angle (deg): 1.1424
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
3400
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 487.5622
Explosive density is within recommended range
Plate velocity (m/s): 67.7908
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.1238e+08
Specific volume of explosive after detonation (m^3/kg): 0.0020
28
7. Min dynamic angle (deg): 0.5629
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6900
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.3582e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 67.7916
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 6.3532e+08
Specific volume of explosive after detonation (m^3/kg): 7.2903e-04
8. Min dynamic angle (deg): 0.4831
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
8040
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.6418e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 67.7917
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 8.9485e+08
Specific volume of explosive after detonation (m^3/kg): 6.0396e-04
9. Min dynamic angle (deg): 0.5179
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6100
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.1592e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 55.1409
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 3.8991e+08
Specific volume of explosive after detonation (m^3/kg): 8.5486e-04
10. Min dynamic angle (deg): 0.9292
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
3400
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 487.5622
Explosive density is within recommended range
Plate velocity (m/s): 55.1405
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 9.1407e+07
Specific volume of explosive after detonation (m^3/kg): 0.0020
11. Min dynamic angle (deg): 0.4579
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6900
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.3582e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 55.1409
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 5.1676e+08
Specific volume of explosive after detonation (m^3/kg): 7.3038e-04
12. Min dynamic angle (deg): 0.3930
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
8040
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.6418e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 55.1410
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 7.2786e+08
Specific volume of explosive after detonation (m^3/kg): 6.0491e-04
29
13. Min dynamic angle (deg): 0.7915
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6100
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 1.1592e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 84.2670
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 5.9586e+08
Specific volume of explosive after detonation (m^3/kg): 8.5074e-04
14. Min dynamic angle (deg): 1.4201
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
3400
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 487.5622
Explosive density is within recommended range
Plate velocity (m/s): 84.2655
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.3969e+08
Specific volume of explosive after detonation (m^3/kg): 0.0020
15. Min dynamic angle (deg): 0.6997
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6900
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.3582e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 84.2672
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 7.8972e+08
Specific volume of explosive after detonation (m^3/kg): 7.2727e-04
16. Min dynamic angle (deg): 0.6005
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
8040
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.6418e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 84.2673
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.1123e+09
Specific volume of explosive after detonation (m^3/kg): 6.0271e-04
30
APPENDIX F
MATLAB Program Output for Surface Condition, C0 = 1.2
1. Min dynamic angle (deg): 1.8149
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6100
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 1.1592e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 193.2103
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.3662e+09
Specific volume of explosive after detonation (m^3/kg): 8.3534e-04
2. Min dynamic angle (deg): 3.2561
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
3400
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 487.5622
Explosive density is within recommended range
Plate velocity (m/s): 193.1924
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 3.2026e+08
Specific volume of explosive after detonation (m^3/kg): 0.0019
3. Min dynamic angle (deg): 1.6044
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6900
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.3582e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 193.2120
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.8107e+09
Specific volume of explosive after detonation (m^3/kg): 7.1565e-04
4. Min dynamic angle (deg): 1.3769
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
8040
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.6418e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 193.2137
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 2.5504e+09
Specific volume of explosive after detonation (m^3/kg): 5.9445e-04
5. Min dynamic angle (deg): 1.2735
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6100
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.1592e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 135.5811
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 9.5871e+08
Specific volume of explosive after detonation (m^3/kg): 8.4349e-04
31
6. Min dynamic angle (deg): 2.2848
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
3400
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 487.5622
Explosive density is within recommended range
Plate velocity (m/s): 135.5749
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 2.2474e+08
Specific volume of explosive after detonation (m^3/kg): 0.0020
7. Min dynamic angle (deg): 1.1259
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6900
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.3582e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 135.5817
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.2706e+09
Specific volume of explosive after detonation (m^3/kg): 7.2180e-04
8. Min dynamic angle (deg): 0.9662
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
8040
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.6418e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 135.5822
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.7897e+09
Specific volume of explosive after detonation (m^3/kg): 5.9882e-04
9. Min dynamic angle (deg): 1.0359
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6100
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.1592e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 110.2807
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 7.7981e+08
Specific volume of explosive after detonation (m^3/kg): 8.4707e-04
10. Min dynamic angle (deg): 1.8584
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
3400
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 487.5622
Explosive density is within recommended range
Plate velocity (m/s): 110.2774
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.8281e+08
Specific volume of explosive after detonation (m^3/kg): 0.0020
11. Min dynamic angle (deg): 0.9158
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6900
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.3582e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 110.2810
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.0335e+09
Specific volume of explosive after detonation (m^3/kg): 7.2450e-04
32
12. Min dynamic angle (deg): 0.7859
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
8040
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.6418e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 110.2813
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.4557e+09
Specific volume of explosive after detonation (m^3/kg): 6.0074e-04
13. Min dynamic angle (deg): 1.5830
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6100
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 1.1592e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 168.5300
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.1917e+09
Specific volume of explosive after detonation (m^3/kg): 8.3883e-04
14. Min dynamic angle (deg): 2.8401
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
3400
Detonation velocity is within recommended range
Density of explosive (kg/m^3): 487.5622
Explosive density is within recommended range
Plate velocity (m/s): 168.5181
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 2.7935e+08
Specific volume of explosive after detonation (m^3/kg): 0.0019
15. Min dynamic angle (deg): 1.3995
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
6900
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.3582e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 168.5312
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 1.5794e+09
Specific volume of explosive after detonation (m^3/kg): 7.1828e-04
16. Min dynamic angle (deg): 1.2010
**NOTE** Min dynamic angle is below recommended range
Detonation velocity (m/s):
8040
**NOTE** Detonation velocity is above recommended range
Density of explosive (kg/m^3): 1.6418e+03
**NOTE** Explosive density is above recommended range
Plate velocity (m/s): 168.5323
**NOTE** Plate velocity is below recommended range
Pressure of shock (N/m^2): 2.2246e+09
Specific volume of explosive after detonation (m^3/kg): 5.9632e-04
33
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Appleton Company, 1912.
[2] Lorenette, G., ”Alfred Nobel,” His Life and Work, The Nobel Foundation, Stockholm,
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[3] U. S. Dept. of Defense, “Military Explosives,” Headquarters, Dept. of the Army, 1990.
[4] Banker, J. G., Reineke, E. G., “ASM Handbook,” Welding, Brazing and Soldering,
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[6] Kudinov, V., Zakhazenko, I., “Criteria for Selecting the Parameters of Explosive
Welding,” Welding Productions, Vol. 32, 1985.
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[8] Commander, Naval Ordnance Systems Command, “Fundamentals of Naval
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[10] Blazynski, T., “Explosive Welding, Forming and Compaction,” Applied Science
Publishers, NY, 1983.
[11] Mader, C., “Numerical Modeling of Explosives and Propellants,” 2nd Ed., CRC
Press, Boca Raton, NY, 1998.
[12] Crossland, B., “Explosive Welding of Metals and its Application,” Clarendon Press,
Oxford, 1982.
[13] Zel’dovich, Y., Raizer, Y., “Physics of Shock Waves and High-Temperature
Hydrodynamic Phenomena,” Vol. 1, Academic Press, NY, 1966.
34
[14] Murr, L. E., et. al., “Novel Deformation Processes and Microstructures
Involving Ballistic Penetrator Formation and Hypervelocity Impact and
Penetration Phenomena," The University of Texas at El Paso, TX, Materials
Characterization 37:245-276, Elsevier Science Inc., 1996.
[15] Petryk, H., et. al., “An Energy Approach to the Formation of Twins in TiAl,”
Metallurgical and Materials Transactions, Vol. 34A, 2003.
[16] Lancaster, J. F., “Metallurgy of Welding,” Chapman & Hall, New York, 1993.
35