# Development of a Numerical Model for an Expanding Tube with Linear Explosive Using AUTODYN

Authors: Mijin Choi; Jung-Ryul Lee; Cheol-Won Kong
Journal: Shock and Vibration
 (2014)

---

## Abstract

Pyrotechnic devices have been employed in satellite launch vehicle missions, generally for the separation of structural subsystems such as stage and satellite separation. Expanding tubes are linear explosives enclosed by an oval steel tube and have been widely used for pyrotechnic joint separation systems. A numerical model is proposed for the prediction of the proper load of an expanding tube using a nonlinear dynamic analysis code, AUTODYN 2D and 3D. To compute a proper core load, numerical models of the open-ended steel tube and mild detonating tube encasing a high explosive were developed and compared with experimental results. 2D and 3D computational results showed good correlation with ballistic test results. The model will provide more flexibility in expanding tube design, leading to economic benefits in the overall expanding tube development procedure.


---

## Body

## 1. Introduction

Pyrotechnic devices are widely used in many space applications. They are used to perform releasing, cutting, pressurization, ignition, switching, and other mechanical work to initiate flight sequences during space missions, such as the separation of subsystems, boosters, fairings, stages, or payload satellites. Such explosive separation devices generate a shock environment that could have a destructive effect on the structure and hardware, especially on electromechanical and optical equipment. The environment is very complex, and studies have revealed that little information is available describing the basic mechanism of shock transmission and predicting shock response. Therefore, improved guidelines for pyrotechnic design, development, and qualification are clearly needed [1–3].Pyrotechnic devices may generally be divided into point sources and line sources. Typical point sources include explosive bolts, separation nuts, pin pullers and pushers, and certain combinations of point sources for low explosive actuation. Typical linear sources include flexible linearly shaped charges, mild detonating fuses, and Super*Zip for high explosive actuation [4, 5].An example of line sources is shown in Figure1. The pyrotechnic device shown is a high-load-carrying separation system that must act without contamination of the payload and is called an expanding tube in this study. The device is one of the greatest shock producers in aerospace separation systems. The detailed components of the expanding tube are also represented in Figure 1, consisting of an MDF (mild detonating fuse), support, and a flattened steel tube. The MDF is a small diameter extruded tube containing a single strand of explosive cord. It is manufactured by filling a tube with explosive material and extruding the tube using a conventional, multiple-die, cold extrusion process [6–9]. Structure separation is accomplished when the MDF is ignited. When an MDF is initiated, the detonation is propagated along the entire length of the linear explosive at a VOD (velocity of detonation) between 6.0 and 7.5 km/s, with very high shock energy. This chemical reaction acts as mechanical loading, which allows the elliptical steel tube to expand mainly in the minor-axis direction and fractures pin joints. Consequently, the expanded tube can separate structures according to its design, without contamination [8]. Since the performance of expanding tubes is mainly related to the explosive load, investigation of the proper explosive load against expanding tube failure and structural damage is important to its design. Even if the expanding tube design has to be verified in ballistic tests, adding a numerical simulation step between the expanding tube sample fabrication and its ballistic test can be a more effective approach for proper explosive load determination. The numerical model can explain the full physics of the phenomena by solving the governing equations that describe the behavior of the system under consideration.Figure 1
A pyro-separation system [9].In spite of the advantages of numerical simulation, only a few numerical and experimental investigations on pyrotechnic devices can be found in the literature. Quidot [10] adopted LSDYNA to simulate a pyrotechnic separation system. A line source comprised an expanding tube with a 1 mm thick stainless steel case, internal synthetic material, and a lead-RDX cord with a 2 g/m RDX charge. Numerical simulations were performed for this device, as shown in Figure 2. However, the proper explosive load, which is the most important factor for real-world applications of expanding tube design, was not addressed under consideration of the detonation wave propagation, the fracture mechanism, and experimental validation.Figure 2
Numerical simulation of expanding tube free expansion [10].In this study, two- and three-dimensional numerical analyses of expanding tube were performed in order to develop a numerical model with high accuracy. The explicit dynamic code AUTODYN is adopted to simulate expansion of the tube and compute a proper core load. AUTODYN is a hydrocode program that is especially suited to solve the interaction problems of different systems of structure, liquid, and gas together, as found in many studies of high explosive simulation [11–15]. To verify the accuracy of the developed model, the numerical results were compared to experimental ballistic test results for different explosive load conditions.


## 2. Ballistic Tests of Expanding Tubes

Experiments were performed to properly design a pyrotechnic device. Explosive tests were carried out for different core loads of 15, 20, 25, and 30 gr/ft. The expanding tube is an open-ended cylinder, in which an MDF filled with hexanitrostilbene (HNS) explosive is set along the central axis of the cylinder. All MDF cords were manufactured with 3 mm diameters regardless of the respective explosive loads, and the wall thickness of the sheath was 0.1 mm. Figure3 represents a section view of an expanding tube with a major axis of 20 mm, a minor axis of 8 mm, and a thickness of 1 mm, regardless of the core load. The tests were carried out 4 or 5 times for each core case, and the deformed shape and failure were investigated. Figure 4 shows the results of ballistic tests with explosive core loads of 15, 20, 25, and 30 gr/ft. Expanding tubes with 15, 20, and 25 gr/ft were freely expanded without failure. In contrast, the 30 gr/ft expanding tube failed at both ends of the cylinder, as shown in Figure 4. The dimensions of each expanding tube were measured, and these measurements were averaged. These results are used to validate the numerical model, which will be discussed in Section 3.Figure 3
Geometry of expanding tube.Figure 4
Experimental results of expanding tube.It can be deduced from the ballistic test results that the threshold explosive load of the tested samples must lie between 25 gr/ft and 30 gr/ft. Ballistic tests are a good approach to determine a proper explosive load for expanding tubes, as well as a better approximation of actual conditions. However, this experimental approach has several disadvantages, in which ballistic tests typically require many expanding tube samples and connected structures, which are also damaged during the test and are nonreusable. The experimental method is also not flexible for design change and is expensive and time consuming. Moreover, it is difficult to explain the physical phenomena of structure and detonation interactions. Therefore, to mitigate this problem, the ballistic test should be the final confirmation test of the expanding tube development procedure, after replacing the numerous case samples with a reliable numerical model.


## 3. Numerical Modeling

### 3.1. Numerical Model

Numerical simulations were performed for all the experiments using hydrocode in AUTODYN 2D, examining the experimental expanding behavior and numerical stresses. The two-dimensional model was made of a steel tube, support, MDF, and surrounding air, as shown in Figure5. The surrounding air was modeled as an ideal gas, and the area was 400 mm × 400 mm. Since the material of the support layer was Teflon and it did not affect tube expansion significantly, it was assumed as air. The explosive material was HNS 1.65, where 1.65 indicates the density of HNS which is 1.65 g/cm3. The explosive loads for all cases ranged from 15 to 30 gr/ft. The explosive propagation velocity for HNS 1.65 was approximately 7,030 m/s, and the detonation point was set to the center of the MDF.Figure 5
Material location of two-dimensional expanding tube.


### 3.2. Material Description and Boundary Conditions

Both the reacted solid and reacted gaseous products of HNS 1.65 explosive were characterized with the Jones-Wilkins-Lee equation of state (JWL EOS, [16]). The pressure in either phase is defined in terms of volume and internal energy independent of temperature as
(1)
p
=
A
(
1
-
ω
R
1
V
)
e
-
R
1
V
+
B
(
1
-
ω
R
2
V
)
e
-
R
2
V
+
ω
e
V
,
where V
=
ρ
/
ρ
o is the relative volume, e is the internal energy, and A, B, R
1, R
2, and ω are empirically determined constants that depend on the kind of the explosive. Table 1 lists the JWL EOS coefficients used for this analysis of the HNS with a density of 1.65 g/cm3.Table 1
Material coefficients in Jones-Wilkins-Lee (JWL) equation for HNS 1.65.
Parameter
Value
Unit
A
4
.
65
×
10
8
kPa
B
8.87
×
10
6
kPa
R
1
4.55
—
R
2
1.35
—
ω
0.35
—The ideal gas EOS for air is shown as follows:(2)
p
=
(
γ
-
1
)
ρ
e
,
where γ is the ideal gas constant and ρ and e are the density and internal energy of the air. The internal energy of air at room temperature and ambient pressure is 2.068 × 105 kJ.Both the tube and sheath were modeled as SS304 steel and AL6061. With the high pressure resulting from explosive detonation, both the tube and sheath undergoing high strain rates were modeled using the shock equation of state [17] and Steinberg-Guinan strength model [18]. Both the von Mises yield strength (Y) and the shear modulus (G) of the Steinberg-Guinan model are assumed to be described as a function of pressure (P), density (ρ), and temperature (T), and, in the case of yield strength, the effective plastic strain as well (ε):
(3)
G
=
G
0
(
1
+
(
G
0
′
G
0
)
p
η
1
/
3
+
(
G
T
′
G
0
)
(
T
-
300
)
)
,
Y
=
Y
0
{
1
+
β
(
ε
+
ε
i
)
}
n
×
(
1
+
(
Y
p
′
Y
0
)
p
η
1
/
3
+
(
Y
T
′
Y
0
)
(
T
-
300
)
)
,
where β is the hardening constant and n is the hardening exponent. The subscript 0 refers to the reference state (T
=
300 K, P
=
0, ε
=
0). The primed parameters with the subscripts P and T indicate the derivatives of the parameters with respect to pressure or temperature at the reference state. Table 2 summarizes the Steinberg-Guinan strength parameters used for the numerical model for the SS304 steel and AL6061 used in the present analysis.Table 2
Steinberg-Guinan strength parameters.
Parameter
SS304
AL6061
Unit
G
7
.
7
×
10
8
2.97
×
10
7
kPa
Y
3.4
×
10
5
5
×
10
4
kPa
β
43
28
—
n
0.35
0.8
—
d
G
/
d
P
1.74
1.4
d
G
/
d
T
-
3.5
×
10
4
-
1.29
×
10
4
kPa/K
d
Y
/
d
P
7.68
×
10
-
3
2.35
×
10
-
3
—
Melting temperature
2380
1604
KA flow-out condition was applied to all boundaries of the Euler grid. This allowed the material to freely leave the grid and also prevented relief waves from being generated off the lateral boundaries of the solid, which would reduce the overall strength of the compressive shock and rarefaction waves.


### 3.3. Failure Model

Spallation (spall-fracture) is a kind of fracture that occurs as planar separation of material parallel to the incident plane wave fronts as a result of dynamic tensile stress components perpendicular to this plane [19]. The spall strength for tensile stress generated when a rarefaction wave travels through material is 2.1 GPa for the steel tube. If the tensile stress of the tube exceeds the predefined spall strength, the material is considered to have “failed.” The tensile stress produced by a rarefaction wave is often shown as a negative pressure [14].


### 3.4. Solvers

AUTODYN provides various solvers for nonlinear problems [11, 20, 21]. Lagrange and Euler solvers were adopted in these FSI (fluid-structure interaction) simulations. The Lagrangian representation keeps material in its initial element, and the numerical mesh moves and deforms with the material. The Lagrange method is most appropriate for the description of solid-like structures. Eulerian representation allows materials to flow from cell to cell, while the mesh is spatially fixed. The Eulerian method is typically well suited for representing fluids and gases [20–23]. In this study, the steel tube and sheath were modeled using a Lagrange solver, while HNS and air were modeled using an Euler solver. The Euler and Lagrange interaction was also implemented to take into account the interaction between fluid (explosive and air) and structures (tube and sheath). This coupling technique allows complex fluid-structure interaction problems that include large deformation of the structure to be solved in a single numerical analysis.


## 3.1. Numerical Model

Numerical simulations were performed for all the experiments using hydrocode in AUTODYN 2D, examining the experimental expanding behavior and numerical stresses. The two-dimensional model was made of a steel tube, support, MDF, and surrounding air, as shown in Figure5. The surrounding air was modeled as an ideal gas, and the area was 400 mm × 400 mm. Since the material of the support layer was Teflon and it did not affect tube expansion significantly, it was assumed as air. The explosive material was HNS 1.65, where 1.65 indicates the density of HNS which is 1.65 g/cm3. The explosive loads for all cases ranged from 15 to 30 gr/ft. The explosive propagation velocity for HNS 1.65 was approximately 7,030 m/s, and the detonation point was set to the center of the MDF.Figure 5
Material location of two-dimensional expanding tube.


## 3.2. Material Description and Boundary Conditions

Both the reacted solid and reacted gaseous products of HNS 1.65 explosive were characterized with the Jones-Wilkins-Lee equation of state (JWL EOS, [16]). The pressure in either phase is defined in terms of volume and internal energy independent of temperature as
(1)
p
=
A
(
1
-
ω
R
1
V
)
e
-
R
1
V
+
B
(
1
-
ω
R
2
V
)
e
-
R
2
V
+
ω
e
V
,
where V
=
ρ
/
ρ
o is the relative volume, e is the internal energy, and A, B, R
1, R
2, and ω are empirically determined constants that depend on the kind of the explosive. Table 1 lists the JWL EOS coefficients used for this analysis of the HNS with a density of 1.65 g/cm3.Table 1
Material coefficients in Jones-Wilkins-Lee (JWL) equation for HNS 1.65.
Parameter
Value
Unit
A
4
.
65
×
10
8
kPa
B
8.87
×
10
6
kPa
R
1
4.55
—
R
2
1.35
—
ω
0.35
—The ideal gas EOS for air is shown as follows:(2)
p
=
(
γ
-
1
)
ρ
e
,
where γ is the ideal gas constant and ρ and e are the density and internal energy of the air. The internal energy of air at room temperature and ambient pressure is 2.068 × 105 kJ.Both the tube and sheath were modeled as SS304 steel and AL6061. With the high pressure resulting from explosive detonation, both the tube and sheath undergoing high strain rates were modeled using the shock equation of state [17] and Steinberg-Guinan strength model [18]. Both the von Mises yield strength (Y) and the shear modulus (G) of the Steinberg-Guinan model are assumed to be described as a function of pressure (P), density (ρ), and temperature (T), and, in the case of yield strength, the effective plastic strain as well (ε):
(3)
G
=
G
0
(
1
+
(
G
0
′
G
0
)
p
η
1
/
3
+
(
G
T
′
G
0
)
(
T
-
300
)
)
,
Y
=
Y
0
{
1
+
β
(
ε
+
ε
i
)
}
n
×
(
1
+
(
Y
p
′
Y
0
)
p
η
1
/
3
+
(
Y
T
′
Y
0
)
(
T
-
300
)
)
,
where β is the hardening constant and n is the hardening exponent. The subscript 0 refers to the reference state (T
=
300 K, P
=
0, ε
=
0). The primed parameters with the subscripts P and T indicate the derivatives of the parameters with respect to pressure or temperature at the reference state. Table 2 summarizes the Steinberg-Guinan strength parameters used for the numerical model for the SS304 steel and AL6061 used in the present analysis.Table 2
Steinberg-Guinan strength parameters.
Parameter
SS304
AL6061
Unit
G
7
.
7
×
10
8
2.97
×
10
7
kPa
Y
3.4
×
10
5
5
×
10
4
kPa
β
43
28
—
n
0.35
0.8
—
d
G
/
d
P
1.74
1.4
d
G
/
d
T
-
3.5
×
10
4
-
1.29
×
10
4
kPa/K
d
Y
/
d
P
7.68
×
10
-
3
2.35
×
10
-
3
—
Melting temperature
2380
1604
KA flow-out condition was applied to all boundaries of the Euler grid. This allowed the material to freely leave the grid and also prevented relief waves from being generated off the lateral boundaries of the solid, which would reduce the overall strength of the compressive shock and rarefaction waves.


## 3.3. Failure Model

Spallation (spall-fracture) is a kind of fracture that occurs as planar separation of material parallel to the incident plane wave fronts as a result of dynamic tensile stress components perpendicular to this plane [19]. The spall strength for tensile stress generated when a rarefaction wave travels through material is 2.1 GPa for the steel tube. If the tensile stress of the tube exceeds the predefined spall strength, the material is considered to have “failed.” The tensile stress produced by a rarefaction wave is often shown as a negative pressure [14].


## 3.4. Solvers

AUTODYN provides various solvers for nonlinear problems [11, 20, 21]. Lagrange and Euler solvers were adopted in these FSI (fluid-structure interaction) simulations. The Lagrangian representation keeps material in its initial element, and the numerical mesh moves and deforms with the material. The Lagrange method is most appropriate for the description of solid-like structures. Eulerian representation allows materials to flow from cell to cell, while the mesh is spatially fixed. The Eulerian method is typically well suited for representing fluids and gases [20–23]. In this study, the steel tube and sheath were modeled using a Lagrange solver, while HNS and air were modeled using an Euler solver. The Euler and Lagrange interaction was also implemented to take into account the interaction between fluid (explosive and air) and structures (tube and sheath). This coupling technique allows complex fluid-structure interaction problems that include large deformation of the structure to be solved in a single numerical analysis.


## 4. Results and Discussion

### 4.1. Two-Dimensional Numerical Results

Figure6 shows the two-dimensional numerical model of an expanding tube explosion. The cell width was approximately 0.16 mm. The number of elements in the tube model was 1,404, and the whole model had 161,504 elements. A description of the numerical model is shown in Table 3.Table 3
Description of numerical model of 2D expanding tube.
Part
Materials
Number of mesh
Solver
Tube
SS304
1,404
Lagrangian
Sheath
AL6061
100
Air
Air
160,000
Eulerian
High explosive
HNS 1.65Figure 6
Two-dimensional expanding tube numerical model.To illustrate the expansion process of the steel tube over time, the deformed configurations of a 15 gr/ft expanding tube are depicted in Figure7 at t
=
0 ms, 0.02 ms, 0.07 ms, and 0.1 ms. The calculation was started at time t
=
0.0 ms and finished at t
=
0.1 ms, when it could be assumed that detonation had been completed. It is clearly seen that the tube is expanded outward radially from the detonation point. At t
=
0.02 ms, early in the detonation, the minor axis of the tube started to be expanded, and the length of the major axis was becoming shorter over time. At a later time, the shape of the tube section was deformed from elliptical to circular, such that the shape change might separate a joint.Figure 7
Expanding tube distortions (explosive load 15 gr/ft).A series of computations was performed by varying explosive loads to compute the material response of the tube, and the deformations are presented in Figure8. It is apparent that the tube expansion increases as the explosive load increases. For a pressure tube in the plastic deformation regime, catastrophic failure is associated with ductile tearing or plastic instability [19, 24, 25]. Rapidly expanding regions (the minor axis of the tube) are accompanied by rapid loss of stress-carrying capability and intense heating due to the dissipation of plastic work, and present risks of rupture [26]. As shown in Figure 8, the plastic region and computed stress were increased proportionally to the explosive load. As a result, the 30 gr/ft expanding tube shows the largest plastic region, and the greatest stress is on the minor axis due to energy absorption. However, the 30 gr/ft expanding tube expanded without spallation in the 2D simulation, which differs distinctly from the experimental results, because this FE analysis accounts for only radial expansion. This suggests that the tube expansion due to the detonation wave propagation should be considered to develop a more reliable numerical model.Figure 8
Material status of expanded steel tube.Figure9 compares the changes of width and height of the tube in the FE analysis and experimental results for three cases, with 15, 20, and 25 gr/ft. Although the calculation results slightly overestimated the experimental results of each core load, the overall error of both width (minor axis) and height (major axis) prediction was less than 10%.Figure 9
Comparison between the experimental and calculational results.


### 4.2. Three-Dimensional Numerical Results

Figure10 illustrates a three-dimensional expanding tube and detonation direction. The open-ended steel tube was 300 mm long, and MDF of a 320 mm length of MDF was inserted into the central bore. Its cross section was the same as in the 2D case. Detonation was assumed to be started from the left end of the tube at t
=
0.0 ms, identically to the experimental conditions. To describe the 3D detonation propagation, a cylindrical blast wave was generated and then remapped in the air. The pressure profile of the detonation wave propagation into the tube is presented in Figure 11, and its structure consists of a shock wave and reaction zone, where the explosive is changed into gaseous products at high temperature. These reaction products act as a piston that allows the shock wave to propagate at a constant velocity [27]. Figure 12 shows the numerical model of the 3D expanding tube. A graded grid was applied to the air and HNS 1.65 to save computation time. Table 4 summarizes the description of the 3D numerical model.Table 4
Description of 3D expanding tube numerical model.
Part
Materials
Number of mesh
Solver
Tube
SS304
64,800
Lagrangian
Sheath
AL6061
13,360
Air
Air
203,125
Eulerian
High explosive
HNS 1.65Figure 10
Three-dimensional expanding tube.Figure 11
Pressure loading profile.Three-dimensional model of expanding tube (front view (a) and isotropic view (b)).
(a)
(b)Figure13 displays the expansion behaviors and material status of each expanding tube. Similar to the 2D results, the tubes with explosive loads of 15–25 gr/ft freely expanded, but the 30 gr/ft tube failed due to plastic deformation. As shown in Figure 13, there was spallation at 130 mm from the side of the 30 gr/ft expanding tube at around t
=
0.07 ms. This ductile fracture was caused by the significant amount of plastic deformation during tube expansion. This failure behavior in FE analysis was somewhat different from the experimental failure behavior. In experiments, two failures occurred on the left and right sides of the steel tube, while one bulk failure was found in the numerical simulation. Even if the first spallation could be predicted, it might be inappropriate to simulate a successive multiple-spallation generation mechanism.Material status of one-sided detonation expanding tube ((a): 15 gr/ft, (b): 20 gr/ft, (c): 25 gr/ft, and (d): 30 gr/ft).
(a)
(b)
(c)
(d)Figure14 compares between the numerical simulation and experimental results of each expanding tube deformation. The overall error was less than 3%, and the agreement was improved by considering the tube expansion due to detonation wave propagation.Figure 14
Comparison between the experimental and calculational results.To compute a proper core load, the internal pressure of the expanding wall of the steel tube was numerically calculated for explosive loads of 15, 20, 25, 27, and 30 gr/ft. The three points indicated in Figure15 represent the placement of a gauge for measuring the internal pressure. Figures 16, 17, and 18 plot the time histories of pressure observed by gauges in the middle of the steel tube thickness over time. The first peak was due to the initial shock wave traveling through the tube. The rarefaction wave generated due to free surface interaction is often shown as negative pressure following the first peak, and it produces tensile stress on the tube. The maximum pressure of the respective gauges ranged from −130 to −520 MPa for the various explosive loads. The pressure histories of expanding tubes with loads of 15–25 gr/ft fluctuated and became stable over time. In the FE analysis, the 27 gr/ft expanding tube was able to expand freely without failure, and its maximum pressure was 1.25 times higher than that of the 25 gr/ft expanding tube, which was found to have a proper explosive load in the ballistic test using expanding tube samples. Therefore, we can conclude that the proper explosive load in the experiments was underestimated according to the numerical results. For an explosive load of 30 gr/ft, the internal pressure increased dramatically and reached a value of −520 MPa. The maximum pressure on the steel tube with the 30 gr/ft load was about 2.7 and 2 times higher than those with the 25 gr/ft and 27 gr/ft loads, respectively. This means that, since the explosive load of 30 gr/ft was greater than the failure threshold, spallation occurred in the actual expanding tube.Figure 15
Gauges position.Figure 16
Pressure histories from gauge number 1.Figure 17
Pressure histories from gauge number 2.Figure 18
Pressure histories from gauge number 3.


## 4.1. Two-Dimensional Numerical Results

Figure6 shows the two-dimensional numerical model of an expanding tube explosion. The cell width was approximately 0.16 mm. The number of elements in the tube model was 1,404, and the whole model had 161,504 elements. A description of the numerical model is shown in Table 3.Table 3
Description of numerical model of 2D expanding tube.
Part
Materials
Number of mesh
Solver
Tube
SS304
1,404
Lagrangian
Sheath
AL6061
100
Air
Air
160,000
Eulerian
High explosive
HNS 1.65Figure 6
Two-dimensional expanding tube numerical model.To illustrate the expansion process of the steel tube over time, the deformed configurations of a 15 gr/ft expanding tube are depicted in Figure7 at t
=
0 ms, 0.02 ms, 0.07 ms, and 0.1 ms. The calculation was started at time t
=
0.0 ms and finished at t
=
0.1 ms, when it could be assumed that detonation had been completed. It is clearly seen that the tube is expanded outward radially from the detonation point. At t
=
0.02 ms, early in the detonation, the minor axis of the tube started to be expanded, and the length of the major axis was becoming shorter over time. At a later time, the shape of the tube section was deformed from elliptical to circular, such that the shape change might separate a joint.Figure 7
Expanding tube distortions (explosive load 15 gr/ft).A series of computations was performed by varying explosive loads to compute the material response of the tube, and the deformations are presented in Figure8. It is apparent that the tube expansion increases as the explosive load increases. For a pressure tube in the plastic deformation regime, catastrophic failure is associated with ductile tearing or plastic instability [19, 24, 25]. Rapidly expanding regions (the minor axis of the tube) are accompanied by rapid loss of stress-carrying capability and intense heating due to the dissipation of plastic work, and present risks of rupture [26]. As shown in Figure 8, the plastic region and computed stress were increased proportionally to the explosive load. As a result, the 30 gr/ft expanding tube shows the largest plastic region, and the greatest stress is on the minor axis due to energy absorption. However, the 30 gr/ft expanding tube expanded without spallation in the 2D simulation, which differs distinctly from the experimental results, because this FE analysis accounts for only radial expansion. This suggests that the tube expansion due to the detonation wave propagation should be considered to develop a more reliable numerical model.Figure 8
Material status of expanded steel tube.Figure9 compares the changes of width and height of the tube in the FE analysis and experimental results for three cases, with 15, 20, and 25 gr/ft. Although the calculation results slightly overestimated the experimental results of each core load, the overall error of both width (minor axis) and height (major axis) prediction was less than 10%.Figure 9
Comparison between the experimental and calculational results.


## 4.2. Three-Dimensional Numerical Results

Figure10 illustrates a three-dimensional expanding tube and detonation direction. The open-ended steel tube was 300 mm long, and MDF of a 320 mm length of MDF was inserted into the central bore. Its cross section was the same as in the 2D case. Detonation was assumed to be started from the left end of the tube at t
=
0.0 ms, identically to the experimental conditions. To describe the 3D detonation propagation, a cylindrical blast wave was generated and then remapped in the air. The pressure profile of the detonation wave propagation into the tube is presented in Figure 11, and its structure consists of a shock wave and reaction zone, where the explosive is changed into gaseous products at high temperature. These reaction products act as a piston that allows the shock wave to propagate at a constant velocity [27]. Figure 12 shows the numerical model of the 3D expanding tube. A graded grid was applied to the air and HNS 1.65 to save computation time. Table 4 summarizes the description of the 3D numerical model.Table 4
Description of 3D expanding tube numerical model.
Part
Materials
Number of mesh
Solver
Tube
SS304
64,800
Lagrangian
Sheath
AL6061
13,360
Air
Air
203,125
Eulerian
High explosive
HNS 1.65Figure 10
Three-dimensional expanding tube.Figure 11
Pressure loading profile.Three-dimensional model of expanding tube (front view (a) and isotropic view (b)).
(a)
(b)Figure13 displays the expansion behaviors and material status of each expanding tube. Similar to the 2D results, the tubes with explosive loads of 15–25 gr/ft freely expanded, but the 30 gr/ft tube failed due to plastic deformation. As shown in Figure 13, there was spallation at 130 mm from the side of the 30 gr/ft expanding tube at around t
=
0.07 ms. This ductile fracture was caused by the significant amount of plastic deformation during tube expansion. This failure behavior in FE analysis was somewhat different from the experimental failure behavior. In experiments, two failures occurred on the left and right sides of the steel tube, while one bulk failure was found in the numerical simulation. Even if the first spallation could be predicted, it might be inappropriate to simulate a successive multiple-spallation generation mechanism.Material status of one-sided detonation expanding tube ((a): 15 gr/ft, (b): 20 gr/ft, (c): 25 gr/ft, and (d): 30 gr/ft).
(a)
(b)
(c)
(d)Figure14 compares between the numerical simulation and experimental results of each expanding tube deformation. The overall error was less than 3%, and the agreement was improved by considering the tube expansion due to detonation wave propagation.Figure 14
Comparison between the experimental and calculational results.To compute a proper core load, the internal pressure of the expanding wall of the steel tube was numerically calculated for explosive loads of 15, 20, 25, 27, and 30 gr/ft. The three points indicated in Figure15 represent the placement of a gauge for measuring the internal pressure. Figures 16, 17, and 18 plot the time histories of pressure observed by gauges in the middle of the steel tube thickness over time. The first peak was due to the initial shock wave traveling through the tube. The rarefaction wave generated due to free surface interaction is often shown as negative pressure following the first peak, and it produces tensile stress on the tube. The maximum pressure of the respective gauges ranged from −130 to −520 MPa for the various explosive loads. The pressure histories of expanding tubes with loads of 15–25 gr/ft fluctuated and became stable over time. In the FE analysis, the 27 gr/ft expanding tube was able to expand freely without failure, and its maximum pressure was 1.25 times higher than that of the 25 gr/ft expanding tube, which was found to have a proper explosive load in the ballistic test using expanding tube samples. Therefore, we can conclude that the proper explosive load in the experiments was underestimated according to the numerical results. For an explosive load of 30 gr/ft, the internal pressure increased dramatically and reached a value of −520 MPa. The maximum pressure on the steel tube with the 30 gr/ft load was about 2.7 and 2 times higher than those with the 25 gr/ft and 27 gr/ft loads, respectively. This means that, since the explosive load of 30 gr/ft was greater than the failure threshold, spallation occurred in the actual expanding tube.Figure 15
Gauges position.Figure 16
Pressure histories from gauge number 1.Figure 17
Pressure histories from gauge number 2.Figure 18
Pressure histories from gauge number 3.


## 5. Conclusions

The development of a numerical model has been presented for the free expansion of expanding tubes by MDF detonation in pyro-separation systems using AUTODYN 2D and 3D hydrocode. Numerical simulations were performed for four expanding tube explosive load cases as a verification set for the numerical model, and additional simulation was done to suggest a more effective explosive load.In the 2D model, since the detonation wave propagation could not be considered, the results slightly overestimated the shape of the expanded tube. To improve the accuracy of the results, the 3D model was developed, which allowed for tube expansion due to detonation wave propagation. The developed model was verified with explosive test results and showed less than 3% overall error. From the results of 3D numerical simulation, where the Steinberg model was applied for the failure criteria of the steel tube, 27 gr/ft was the proper explosive load, and the 30 gr/ft explosive load exceeded the threshold of expanding tube failure. Since the numerical model verified with ballistic test results allows for various design changes, the development cost and time can be greatly reduced by minimizing the number of experimental tests required. Therefore, it is suggested to add a numerical model development step using 3D hydrocode to the overall procedure of expanding tube development for pyrotechnic separation systems.



---

*Source: 436156-2014-03-24.xml*