Micromechanics-based Determination of Effective Elastic Properties of Polymer Bonded Explosives
Biswajit Banerjee and Daniel O. Adams
Dept. of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112
Polymer Bonded Explosives
Polymer bonded explosives are particulate composites. The two primary components
of these composites are explosive particles and a rubbery binder. The volume of
particles in the composite is typically around 90% of the total volume.
Table 1 Typical polymer bonded explosives.
0.87
0.92
0.90
KEL-F-3700
Estane/BDNPA-F
KEL-F-800
Pressed PBX 9501 based microstructures
0.13
0.08
0.10
Modeling Particulate Composites
PBX 9501 contains 92% by volume of HMX (high melting explosive) particles and 8%
by volume of binder. The HMX particles are monoclinic and linear elastic. The binder
is a 1:1 mixture of the rubber Estane 5703 and a plasticizer (BDNPA/F). The
mechanical behavior of the binder is strain rate and temperature dependent. As a
result, the response of PBX 9501 also depends on strain rate and temperature.
V ol. Frac. = 0.1
17.7
0.32
4
10
Eeff /E binder
15.3
0.21
V ol. Frac. = 0.3
DEM
5
10
Poisson's ratio
Expt. MD Simulation
V ol. Frac. = 0.2
5
10
FE M
DE M
PB X 9501 (DE M)
PB X 9501 (E xpt.)
10
Figure 1 Microstructure of PBX 9501.
3
2
1
25 C
22oC
16oC
0oC
o
-15 C
-20oC
-40oC
10
1
10
8
6
V ol. Frac. = 0.92
Figure 9 Models microstructures representing PBX 9501.
0
0.2
0.4
0.6
Particle V ol. Frac.
0.8
1
0
RCM-GMC 16x16
10
6000
5000
4
10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Particle V ol. Frac.
1
8000
FE M
GMC
R CM-GMC
7000
4000
3
10
Pressed PBX 9501
FE M
GMC
R CM-GMC
5000
4000
3000
2
10
3000
2000
1
10
10
PBX 9501 dry blend
6000
2000
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Particle V ol. Frac.
1
1000
0
Figure 6 Differential effective medium (DEM), Recursive Cell Method with FEM homogenizer (RCM-FEM), and Recursive Cell Method with GMC homogenizer (RCM-GMC) vs. finite element estimates.
55oC
o
40 C
27oC
25oC
o
17 C
0oC
o
-20 C
-40oC
-55oC
12
Y oung's Modulus (GPa)
Y oung's Modulus (MPa)
2
14
o
10
PBX 9501
V ol. Frac. = 0.8
0
0
0
10
Contrast = 100000
FE M
R CM
Contrast = 10000
FE M
R CM
Contrast = 1000
FE M
R CM
Contrast = 100
FE M
R CM
Contrast = 10
FE M
R CM
4
10
Binder
V ol. Frac. = 0.7
5
10
1
10
RCM-FEM 16x16
10
10
3
V ol. Frac. = 0.6
10
3
2
10
V ol. Frac. = 0.5
10
10
4
V ol. Frac. = 0.4
Figure 5 Model particulate composite microstructures.
Table 2 Elastic properties of HMX.
Young's modulus (GPa)
Expt.
MD Simulation
Figure 4 Schematic of recursive cell method (RCM).
C 11 (MPa)
RDX
HMX
TATB
Vol. Frac
C 11 (MPa)
PBX 9010
PBX 9501
PBX 9502
Material
Binder
PBX 9501 dry blend based microstructures
E ffective Y oung's Modulus (MPa)
Particles
Material Vol. Frac
E ffective Y oung's Modulus (MPa)
PBX
Modeling PBX 9501
In the recursive cell method (RCM) the representative
volume element (RVE) is divided into a regular
grid of subcells. Instead of the whole RVE,
small square blocks of subcells are homogenized
at a time. The procedure is repeated recursively
until a single homogeneous material remains.
This material is the effective material.
1000
1
2
Model
3
4
0
1
2
Model
3
4
Figure 10 Effective properties of model RVEs from FEM, GMC, RCM-GMC.
Three-D. vs. Two-D:
4
0
10
2
-1
10 -4
10
-2
10
0
10
Strain R ate (/s)
2
10
4
10
0 -4
10
-2
10
0
2
10
Strain R ate (/s)
10
4
10
Figure 7 Three-dimensional models of particulate composites.
Figure 2 Young's modulus of binder and PBX 9501. Poisson's ratio is 0.49 for the binder and 0.35 for PBX 9501.
Glass-Estane composites
Micromechanics
HMX-Binder composites
5
Exact relations and DEM are used to assess the
accuracy of FEM and RCM Bounds can be
observed to be widely separated. Direct GMC
does not model stress bridging accurately.
4
10
Eeff /E binder
Exact Relations
Third-Order Bounds
Differential Effective Medium (DEM)
Finite Element (FEM) Approximations
Generalized Method of Cells (GMC)
Recursive Cell Method (RCM)
10
Figure 11 Finite element prediction vs. experimental data
on the Young's modulus of PBX 9501.
Upper B ound
L ower B ound
PB X 9501 Upper B ound
PB X 9501 L ower B ound
PB X 9501 E xpt.
Conclusions
3
10
2
10
1
10
0
10
0
0.2
0.4
0.6
Particle V ol. Frac.
0.8
1
Figure 3 Third-order bounds and PBX 9501.
Figure 8 Three-dimensional vs. two-dimensional finite element estimates.
-- Two-dimensional finite element estimates are close to differential effective medium predictions.
-- RCM-GMC estimates closer to finite element estimates than RCM-FEM predictions.
-- Difference between two- and three-dimensional finite element estimates is small.
- Predicted effective properties are strongly microstructure dependent.
- Detailed numerical simulations are required for accurate estimates.
- Recursive cell method with the generalized method of cells homogenizer can predict
effective moduli that are close to finite element estimates with considerably greater
computational efficiency.
Acknowledgements
Funded by the Department of Energy grant DE-FG03-02ER45914 and the Center for the Simulation of
Accidental Fires and Explosions, University of Utah, Salt Lake City, UT 84112.